| L(s) = 1 | − 2·5-s − 2·9-s + 8·17-s − 25-s + 4·29-s − 8·37-s − 4·41-s + 4·45-s − 2·49-s + 16·53-s + 20·61-s + 8·73-s − 5·81-s − 16·85-s + 4·89-s + 8·97-s + 4·101-s + 4·109-s + 10·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2/3·9-s + 1.94·17-s − 1/5·25-s + 0.742·29-s − 1.31·37-s − 0.624·41-s + 0.596·45-s − 2/7·49-s + 2.19·53-s + 2.56·61-s + 0.936·73-s − 5/9·81-s − 1.73·85-s + 0.423·89-s + 0.812·97-s + 0.398·101-s + 0.383·109-s + 0.909·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.174046147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.174046147\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−9.662128076682678935095567165572, −8.898565172594837458823501767124, −8.454370003981634384666445347058, −8.182572226663578743085298004918, −7.60056986906470589121404104579, −7.13179636264210348612558264349, −6.64682522472166131489509070865, −5.83123570356027698868579216182, −5.44053230425425632771986827124, −4.94732929598326896221208375801, −4.06010960915291859433782156511, −3.55250343309156722377070530659, −3.07251264667744428047425937347, −2.09803672262678624050543854786, −0.809626660416299295693665810818,
0.809626660416299295693665810818, 2.09803672262678624050543854786, 3.07251264667744428047425937347, 3.55250343309156722377070530659, 4.06010960915291859433782156511, 4.94732929598326896221208375801, 5.44053230425425632771986827124, 5.83123570356027698868579216182, 6.64682522472166131489509070865, 7.13179636264210348612558264349, 7.60056986906470589121404104579, 8.182572226663578743085298004918, 8.454370003981634384666445347058, 8.898565172594837458823501767124, 9.662128076682678935095567165572
