| L(s) = 1 | + 3-s − 2·9-s − 3·11-s − 13-s − 5·17-s − 6·19-s − 5·27-s + 5·29-s − 2·31-s − 3·33-s − 4·37-s − 39-s + 2·41-s + 10·43-s − 9·47-s − 5·51-s + 6·53-s − 6·57-s − 6·59-s − 12·61-s − 2·67-s − 14·73-s + 79-s + 81-s − 12·83-s + 5·87-s − 2·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.904·11-s − 0.277·13-s − 1.21·17-s − 1.37·19-s − 0.962·27-s + 0.928·29-s − 0.359·31-s − 0.522·33-s − 0.657·37-s − 0.160·39-s + 0.312·41-s + 1.52·43-s − 1.31·47-s − 0.700·51-s + 0.824·53-s − 0.794·57-s − 0.781·59-s − 1.53·61-s − 0.244·67-s − 1.63·73-s + 0.112·79-s + 1/9·81-s − 1.31·83-s + 0.536·87-s − 0.207·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3177506641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3177506641\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.06082487570776, −13.43184617891423, −13.20167034236141, −12.52737145299240, −12.18824299305022, −11.41399262970403, −10.93816105446286, −10.61780219392025, −10.02453127571514, −9.373385298085876, −8.807192033598788, −8.541463443000951, −8.022022272864422, −7.405821522118313, −6.917057501323324, −6.142451549572899, −5.866283632689650, −5.024196744658210, −4.524782411110486, −4.011495036270272, −3.163370814678368, −2.619754060522835, −2.256018120812051, −1.458246260450697, −0.1652262749205528,
0.1652262749205528, 1.458246260450697, 2.256018120812051, 2.619754060522835, 3.163370814678368, 4.011495036270272, 4.524782411110486, 5.024196744658210, 5.866283632689650, 6.142451549572899, 6.917057501323324, 7.405821522118313, 8.022022272864422, 8.541463443000951, 8.807192033598788, 9.373385298085876, 10.02453127571514, 10.61780219392025, 10.93816105446286, 11.41399262970403, 12.18824299305022, 12.52737145299240, 13.20167034236141, 13.43184617891423, 14.06082487570776
