| L(s) = 1 | + 4·3-s − 2·9-s − 12·23-s − 4·25-s − 40·27-s − 12·29-s − 12·41-s − 36·47-s + 28·49-s − 48·69-s − 16·75-s − 55·81-s − 48·87-s + 48·101-s + 4·121-s − 48·123-s + 127-s + 131-s + 137-s + 139-s − 144·141-s + 112·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 2/3·9-s − 2.50·23-s − 4/5·25-s − 7.69·27-s − 2.22·29-s − 1.87·41-s − 5.25·47-s + 4·49-s − 5.77·69-s − 1.84·75-s − 6.11·81-s − 5.14·87-s + 4.77·101-s + 4/11·121-s − 4.32·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 12.1·141-s + 9.23·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3985007068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3985007068\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 47 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 125 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 152 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 164 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.43478046078107329586528432807, −6.36593492147629343623786684738, −6.32191362593348365794404325148, −5.71120169066776123779638387291, −5.65384610217344348169859750497, −5.61847347114298276627142261224, −5.61274604279707611840117975904, −4.98279313822080111030086588460, −4.87528081723599115190879580249, −4.75502408066513450002033873557, −4.15111639098684444103440491684, −3.92790280088325373947609346548, −3.71371550653991960959025391469, −3.68589689846598288322817894372, −3.38263240706867473309418610355, −3.33938658896238229278194786875, −2.85686275816703161421351086564, −2.72366215782119817803986450294, −2.50161683764124524329507681317, −2.02074777555634067416883342830, −2.00231717390234486346829632302, −1.83860759414197422465421681256, −1.54951259656422904729164029608, −0.54267309294496647913256063402, −0.11866920241489012450222362196,
0.11866920241489012450222362196, 0.54267309294496647913256063402, 1.54951259656422904729164029608, 1.83860759414197422465421681256, 2.00231717390234486346829632302, 2.02074777555634067416883342830, 2.50161683764124524329507681317, 2.72366215782119817803986450294, 2.85686275816703161421351086564, 3.33938658896238229278194786875, 3.38263240706867473309418610355, 3.68589689846598288322817894372, 3.71371550653991960959025391469, 3.92790280088325373947609346548, 4.15111639098684444103440491684, 4.75502408066513450002033873557, 4.87528081723599115190879580249, 4.98279313822080111030086588460, 5.61274604279707611840117975904, 5.61847347114298276627142261224, 5.65384610217344348169859750497, 5.71120169066776123779638387291, 6.32191362593348365794404325148, 6.36593492147629343623786684738, 6.43478046078107329586528432807
