| L(s) = 1 | + 1.23e3·3-s + 5.74e4·5-s − 6.42e4·7-s − 6.66e4·9-s + 2.46e6·11-s − 8.03e6·13-s + 7.10e7·15-s + 7.11e7·17-s + 1.36e8·19-s − 7.93e7·21-s + 1.18e9·23-s + 2.07e9·25-s − 2.05e9·27-s + 8.90e8·29-s − 4.59e9·31-s + 3.04e9·33-s − 3.69e9·35-s + 1.95e10·37-s − 9.92e9·39-s − 2.72e9·41-s + 5.17e10·43-s − 3.82e9·45-s + 5.35e10·47-s − 9.27e10·49-s + 8.78e10·51-s − 8.26e10·53-s + 1.41e11·55-s + ⋯ |
| L(s) = 1 | + 0.978·3-s + 1.64·5-s − 0.206·7-s − 0.0417·9-s + 0.419·11-s − 0.461·13-s + 1.60·15-s + 0.714·17-s + 0.664·19-s − 0.201·21-s + 1.67·23-s + 1.70·25-s − 1.01·27-s + 0.278·29-s − 0.930·31-s + 0.410·33-s − 0.339·35-s + 1.25·37-s − 0.451·39-s − 0.0895·41-s + 1.24·43-s − 0.0687·45-s + 0.724·47-s − 0.957·49-s + 0.699·51-s − 0.512·53-s + 0.689·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(4.402344411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.402344411\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 412 p T + p^{13} T^{2} \) |
| 5 | \( 1 - 2298 p^{2} T + p^{13} T^{2} \) |
| 7 | \( 1 + 9176 p T + p^{13} T^{2} \) |
| 11 | \( 1 - 224052 p T + p^{13} T^{2} \) |
| 13 | \( 1 + 8032766 T + p^{13} T^{2} \) |
| 17 | \( 1 - 71112402 T + p^{13} T^{2} \) |
| 19 | \( 1 - 136337060 T + p^{13} T^{2} \) |
| 23 | \( 1 - 1186563144 T + p^{13} T^{2} \) |
| 29 | \( 1 - 890583090 T + p^{13} T^{2} \) |
| 31 | \( 1 + 4595552672 T + p^{13} T^{2} \) |
| 37 | \( 1 - 19585053898 T + p^{13} T^{2} \) |
| 41 | \( 1 + 2724170358 T + p^{13} T^{2} \) |
| 43 | \( 1 - 51762321116 T + p^{13} T^{2} \) |
| 47 | \( 1 - 53572833168 T + p^{13} T^{2} \) |
| 53 | \( 1 + 82633440006 T + p^{13} T^{2} \) |
| 59 | \( 1 + 394266352980 T + p^{13} T^{2} \) |
| 61 | \( 1 + 671061772142 T + p^{13} T^{2} \) |
| 67 | \( 1 - 388156449812 T + p^{13} T^{2} \) |
| 71 | \( 1 - 388772243928 T + p^{13} T^{2} \) |
| 73 | \( 1 - 1540972938026 T + p^{13} T^{2} \) |
| 79 | \( 1 - 3306509559280 T + p^{13} T^{2} \) |
| 83 | \( 1 - 4931756967396 T + p^{13} T^{2} \) |
| 89 | \( 1 - 3502949738490 T + p^{13} T^{2} \) |
| 97 | \( 1 + 388932598558 T + p^{13} T^{2} \) |
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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
−12.53156404789902592567953850079, −10.86570870903117494316334226301, −9.458134450580845502690838067804, −9.234016751518337128718376581237, −7.64165373072647504707728513314, −6.25140787629370610388115395175, −5.13615972131002144203086377213, −3.24207120031939686381014085709, −2.33476827978818215593591584320, −1.13635690918101706536048702891,
1.13635690918101706536048702891, 2.33476827978818215593591584320, 3.24207120031939686381014085709, 5.13615972131002144203086377213, 6.25140787629370610388115395175, 7.64165373072647504707728513314, 9.234016751518337128718376581237, 9.458134450580845502690838067804, 10.86570870903117494316334226301, 12.53156404789902592567953850079
