| L(s) = 1 | − 8·5-s + 4·13-s + 4·17-s + 38·25-s + 16·29-s + 20·37-s − 8·41-s + 28·53-s − 32·65-s + 28·73-s + 18·81-s − 32·85-s − 28·97-s + 48·109-s + 36·113-s − 20·121-s − 136·125-s + 127-s + 131-s + 137-s + 139-s − 128·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3.57·5-s + 1.10·13-s + 0.970·17-s + 38/5·25-s + 2.97·29-s + 3.28·37-s − 1.24·41-s + 3.84·53-s − 3.96·65-s + 3.27·73-s + 2·81-s − 3.47·85-s − 2.84·97-s + 4.59·109-s + 3.38·113-s − 1.81·121-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.6·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.460927437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.460927437\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 542 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 2702 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 3326 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 2578 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 2606 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.20924957123185824353214197929, −6.58325554751861152459748669912, −6.54924465860822829057565795998, −6.34242811483932212847056428215, −6.20046281201546766898151929982, −5.87858510084929993875931566238, −5.49417876278610238580917008599, −5.22749700962056678580572602532, −4.99218500360125379989454933050, −4.82787669480671824358480454860, −4.58329646297923248248155734147, −4.29358123229920425813289483807, −4.15252359049748957888910733822, −3.93090077919749968533425366975, −3.55991014931965785906009800304, −3.52288164616723216047463528685, −3.42233251412544790252163464322, −2.86512113557611190551058533997, −2.67356148997737691957147354396, −2.53742082632452432688807137161, −2.05601865727651715919998004771, −1.31482852011629183228314243358, −0.844412216663648797129275997289, −0.830057158570300643312637949553, −0.57071229291142128264431326086,
0.57071229291142128264431326086, 0.830057158570300643312637949553, 0.844412216663648797129275997289, 1.31482852011629183228314243358, 2.05601865727651715919998004771, 2.53742082632452432688807137161, 2.67356148997737691957147354396, 2.86512113557611190551058533997, 3.42233251412544790252163464322, 3.52288164616723216047463528685, 3.55991014931965785906009800304, 3.93090077919749968533425366975, 4.15252359049748957888910733822, 4.29358123229920425813289483807, 4.58329646297923248248155734147, 4.82787669480671824358480454860, 4.99218500360125379989454933050, 5.22749700962056678580572602532, 5.49417876278610238580917008599, 5.87858510084929993875931566238, 6.20046281201546766898151929982, 6.34242811483932212847056428215, 6.54924465860822829057565795998, 6.58325554751861152459748669912, 7.20924957123185824353214197929
