| L(s) = 1 | − 1.23e12·2-s − 2.24e18·3-s + 9.23e23·4-s − 5.14e27·5-s + 2.77e30·6-s + 4.93e32·7-s − 3.94e35·8-s − 4.42e37·9-s + 6.36e39·10-s + 1.90e41·11-s − 2.07e42·12-s − 7.98e43·13-s − 6.09e44·14-s + 1.15e46·15-s − 7.02e46·16-s − 4.22e48·17-s + 5.46e49·18-s − 5.43e50·19-s − 4.75e51·20-s − 1.10e51·21-s − 2.35e53·22-s − 6.81e53·23-s + 8.86e53·24-s + 9.94e54·25-s + 9.87e55·26-s + 2.09e56·27-s + 4.55e56·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.319·3-s + 1.52·4-s − 1.26·5-s + 0.508·6-s + 0.204·7-s − 0.840·8-s − 0.897·9-s + 2.01·10-s + 1.39·11-s − 0.488·12-s − 0.796·13-s − 0.325·14-s + 0.404·15-s − 0.192·16-s − 1.05·17-s + 1.42·18-s − 1.67·19-s − 1.93·20-s − 0.0655·21-s − 2.21·22-s − 1.10·23-s + 0.268·24-s + 0.601·25-s + 1.26·26-s + 0.606·27-s + 0.313·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(80-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+79/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(40)\) |
\(\approx\) |
\(0.1206329045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1206329045\) |
| \(L(\frac{81}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.23e12T + 6.04e23T^{2} \) |
| 3 | \( 1 + 2.24e18T + 4.92e37T^{2} \) |
| 5 | \( 1 + 5.14e27T + 1.65e55T^{2} \) |
| 7 | \( 1 - 4.93e32T + 5.79e66T^{2} \) |
| 11 | \( 1 - 1.90e41T + 1.86e82T^{2} \) |
| 13 | \( 1 + 7.98e43T + 1.00e88T^{2} \) |
| 17 | \( 1 + 4.22e48T + 1.60e97T^{2} \) |
| 19 | \( 1 + 5.43e50T + 1.05e101T^{2} \) |
| 23 | \( 1 + 6.81e53T + 3.77e107T^{2} \) |
| 29 | \( 1 + 1.01e58T + 3.38e115T^{2} \) |
| 31 | \( 1 - 1.25e58T + 6.57e117T^{2} \) |
| 37 | \( 1 + 6.27e61T + 7.72e123T^{2} \) |
| 41 | \( 1 - 6.35e63T + 2.56e127T^{2} \) |
| 43 | \( 1 + 1.89e64T + 1.10e129T^{2} \) |
| 47 | \( 1 + 6.42e65T + 1.24e132T^{2} \) |
| 53 | \( 1 - 1.33e68T + 1.65e136T^{2} \) |
| 59 | \( 1 + 2.01e69T + 7.89e139T^{2} \) |
| 61 | \( 1 - 1.94e69T + 1.09e141T^{2} \) |
| 67 | \( 1 + 1.81e72T + 1.81e144T^{2} \) |
| 71 | \( 1 - 1.18e73T + 1.77e146T^{2} \) |
| 73 | \( 1 - 2.44e73T + 1.59e147T^{2} \) |
| 79 | \( 1 - 1.60e74T + 8.17e149T^{2} \) |
| 83 | \( 1 - 4.26e75T + 4.04e151T^{2} \) |
| 89 | \( 1 - 1.07e77T + 1.00e154T^{2} \) |
| 97 | \( 1 + 3.66e78T + 9.01e156T^{2} \) |
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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
−16.57379500823388087917171666340, −14.90927712438627648979983198492, −11.86862945983805402032115812380, −10.96810279300350721435114564975, −9.089113819861902983881996800027, −8.014057266102467178760383705238, −6.61145025976504718644567543284, −4.13560599012070053392534113422, −1.99966424155203966989268386725, −0.25326550789926601380635987089,
0.25326550789926601380635987089, 1.99966424155203966989268386725, 4.13560599012070053392534113422, 6.61145025976504718644567543284, 8.014057266102467178760383705238, 9.089113819861902983881996800027, 10.96810279300350721435114564975, 11.86862945983805402032115812380, 14.90927712438627648979983198492, 16.57379500823388087917171666340
