| L(s) = 1 | + 2-s − 4-s − 3·8-s − 9-s + 2·11-s − 16-s − 18-s + 2·22-s + 2·23-s − 4·25-s − 8·29-s + 5·32-s + 36-s − 2·44-s + 2·46-s − 4·50-s + 8·53-s − 8·58-s + 7·64-s − 12·67-s − 18·71-s + 3·72-s − 4·79-s + 81-s − 6·88-s − 2·92-s − 2·99-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1/3·9-s + 0.603·11-s − 1/4·16-s − 0.235·18-s + 0.426·22-s + 0.417·23-s − 4/5·25-s − 1.48·29-s + 0.883·32-s + 1/6·36-s − 0.301·44-s + 0.294·46-s − 0.565·50-s + 1.09·53-s − 1.05·58-s + 7/8·64-s − 1.46·67-s − 2.13·71-s + 0.353·72-s − 0.450·79-s + 1/9·81-s − 0.639·88-s − 0.208·92-s − 0.201·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 156 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.739669822998969445809140863235, −8.034516887502443310649335442709, −7.52685283860693278740387088025, −7.15143906922658593640837830289, −6.45585929004488164739050655816, −5.99444197952005469662508385604, −5.64425540196005984867157019856, −5.16050516489234500565730154860, −4.52358344101759891331149131012, −4.06963482182445630773446579280, −3.58887616485273778765261206279, −3.02437764916907353123409236856, −2.28485132977309640466176859544, −1.33358386294238772488720373779, 0,
1.33358386294238772488720373779, 2.28485132977309640466176859544, 3.02437764916907353123409236856, 3.58887616485273778765261206279, 4.06963482182445630773446579280, 4.52358344101759891331149131012, 5.16050516489234500565730154860, 5.64425540196005984867157019856, 5.99444197952005469662508385604, 6.45585929004488164739050655816, 7.15143906922658593640837830289, 7.52685283860693278740387088025, 8.034516887502443310649335442709, 8.739669822998969445809140863235
