| L(s) = 1 | + 12·13-s + 8·25-s − 24·37-s + 14·49-s + 24·61-s + 32·73-s + 16·97-s − 12·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 3.32·13-s + 8/5·25-s − 3.94·37-s + 2·49-s + 3.07·61-s + 3.74·73-s + 1.62·97-s − 1.14·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.482674353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.482674353\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−8.991704092772574245798637408766, −8.665448327469721039064885177842, −8.520252027462985877460482464529, −8.358302792435375332841741510610, −7.68150653352053951624202895369, −7.18987703843219766734943236132, −6.71844592878880321149052682834, −6.53037306920614621635227014891, −6.26246697919264591378252245852, −5.50993757594403209479604473140, −5.25931563034502464343249746713, −5.10944623049805629349569752524, −4.06305588673099705790410299508, −3.92845347855090775718477750503, −3.48144293144950964823622845660, −3.22571441015524043697539923249, −2.36249069848300583001755989159, −1.83646560878485289233767997593, −1.16226903652818872137226635763, −0.75773277705540921345974271839,
0.75773277705540921345974271839, 1.16226903652818872137226635763, 1.83646560878485289233767997593, 2.36249069848300583001755989159, 3.22571441015524043697539923249, 3.48144293144950964823622845660, 3.92845347855090775718477750503, 4.06305588673099705790410299508, 5.10944623049805629349569752524, 5.25931563034502464343249746713, 5.50993757594403209479604473140, 6.26246697919264591378252245852, 6.53037306920614621635227014891, 6.71844592878880321149052682834, 7.18987703843219766734943236132, 7.68150653352053951624202895369, 8.358302792435375332841741510610, 8.520252027462985877460482464529, 8.665448327469721039064885177842, 8.991704092772574245798637408766
