| L(s) = 1 | − 8·3-s − 114·9-s − 200·11-s − 540·17-s + 1.17e3·19-s + 574·25-s + 1.68e3·27-s + 1.60e3·33-s − 4.10e3·41-s − 904·43-s − 2.94e3·49-s + 4.32e3·51-s − 9.40e3·57-s − 2.21e3·59-s + 1.22e4·67-s − 7.33e3·73-s − 4.59e3·75-s + 5.41e3·81-s − 2.00e4·83-s + 1.50e3·89-s − 1.30e4·97-s + 2.28e4·99-s + 2.39e4·107-s + 1.65e4·113-s + 718·121-s + 3.28e4·123-s + 127-s + ⋯ |
| L(s) = 1 | − 8/9·3-s − 1.40·9-s − 1.65·11-s − 1.86·17-s + 3.25·19-s + 0.918·25-s + 2.31·27-s + 1.46·33-s − 2.43·41-s − 0.488·43-s − 1.22·49-s + 1.66·51-s − 2.89·57-s − 0.636·59-s + 2.73·67-s − 1.37·73-s − 0.816·75-s + 0.824·81-s − 2.91·83-s + 0.189·89-s − 1.39·97-s + 2.32·99-s + 2.08·107-s + 1.29·113-s + 0.0490·121-s + 2.16·123-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1636805503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1636805503\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + 4 T + p^{4} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 574 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2942 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 100 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 55678 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 270 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 588 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 556546 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 378238 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1673986 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2815166 T^{2} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 50 p T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 452 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7995778 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 7451966 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 1108 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 27314686 T^{2} + p^{8} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 6140 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 20264578 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 3666 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 76136578 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 10036 T + p^{4} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 750 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6542 T + p^{4} T^{2} )^{2} \) |
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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
−11.61840433263332524119671705063, −11.34139853505032077480910401470, −10.80349296659133327311558083907, −10.12234826095503712994965802430, −9.958597364407167005920146529803, −9.058856050330730225881535484490, −8.762340759616639250465240973658, −8.137763153080406586614707779595, −7.78878426048688417306937576047, −6.82531063397824017618749625685, −6.81395123998556797397942604510, −5.84253155350283927469947793306, −5.36921487102043438278993563113, −5.11740867733757738225468579807, −4.66127205477519613905089740964, −3.26548150676730830806383507047, −3.07393365168098654503749209563, −2.33567644061433162481182125495, −1.16570535159592180195158984063, −0.15133330863771436199093731363,
0.15133330863771436199093731363, 1.16570535159592180195158984063, 2.33567644061433162481182125495, 3.07393365168098654503749209563, 3.26548150676730830806383507047, 4.66127205477519613905089740964, 5.11740867733757738225468579807, 5.36921487102043438278993563113, 5.84253155350283927469947793306, 6.81395123998556797397942604510, 6.82531063397824017618749625685, 7.78878426048688417306937576047, 8.137763153080406586614707779595, 8.762340759616639250465240973658, 9.058856050330730225881535484490, 9.958597364407167005920146529803, 10.12234826095503712994965802430, 10.80349296659133327311558083907, 11.34139853505032077480910401470, 11.61840433263332524119671705063
