| L(s) = 1 | + 3·5-s − 3·7-s + 4·11-s − 2·13-s − 3·17-s − 12·19-s + 25-s − 6·31-s − 9·35-s − 7·37-s − 6·41-s − 3·43-s − 9·47-s − 3·49-s + 18·53-s + 12·55-s − 12·59-s + 2·61-s − 6·65-s − 12·67-s + 9·71-s + 20·73-s − 12·77-s − 24·79-s − 10·83-s − 9·85-s + 6·91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s + 1.20·11-s − 0.554·13-s − 0.727·17-s − 2.75·19-s + 1/5·25-s − 1.07·31-s − 1.52·35-s − 1.15·37-s − 0.937·41-s − 0.457·43-s − 1.31·47-s − 3/7·49-s + 2.47·53-s + 1.61·55-s − 1.56·59-s + 0.256·61-s − 0.744·65-s − 1.46·67-s + 1.06·71-s + 2.34·73-s − 1.36·77-s − 2.70·79-s − 1.09·83-s − 0.976·85-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T - 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 76 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 124 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 38 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.359247470924614125970790125691, −8.310080589934033374812375342089, −7.33713268528036076349461049532, −7.09046374063523696582329303791, −6.74069179733242319110287892029, −6.45394742968289184001431464816, −6.16318255530587707759055538034, −5.94728239294354965613847389707, −5.28505144718888428024789098250, −5.08379743035954149505590828294, −4.30570908196938274634522827852, −4.23642977810300404332541306626, −3.58569493556401252252580430744, −3.33471575680941272079501992305, −2.43470077497720609517368412384, −2.38466120869572987838380453888, −1.67036395773389906988686940996, −1.49264770548250293594637876598, 0, 0,
1.49264770548250293594637876598, 1.67036395773389906988686940996, 2.38466120869572987838380453888, 2.43470077497720609517368412384, 3.33471575680941272079501992305, 3.58569493556401252252580430744, 4.23642977810300404332541306626, 4.30570908196938274634522827852, 5.08379743035954149505590828294, 5.28505144718888428024789098250, 5.94728239294354965613847389707, 6.16318255530587707759055538034, 6.45394742968289184001431464816, 6.74069179733242319110287892029, 7.09046374063523696582329303791, 7.33713268528036076349461049532, 8.310080589934033374812375342089, 8.359247470924614125970790125691
