| L(s) = 1 | − 4.60e6·2-s − 1.04e10·3-s + 1.24e13·4-s − 9.11e14·5-s + 4.82e16·6-s + 3.55e17·7-s − 1.68e19·8-s + 1.09e20·9-s + 4.19e21·10-s + 2.29e22·11-s − 1.30e23·12-s − 1.59e24·13-s − 1.63e24·14-s + 9.53e24·15-s − 3.19e25·16-s − 8.94e25·17-s − 5.04e26·18-s − 4.96e27·19-s − 1.13e28·20-s − 3.71e27·21-s − 1.05e29·22-s − 2.67e28·23-s + 1.75e29·24-s − 3.06e29·25-s + 7.36e30·26-s − 1.14e30·27-s + 4.42e30·28-s + ⋯ |
| L(s) = 1 | − 1.55·2-s − 0.577·3-s + 1.41·4-s − 0.854·5-s + 0.897·6-s + 0.240·7-s − 0.644·8-s + 0.333·9-s + 1.32·10-s + 0.936·11-s − 0.816·12-s − 1.79·13-s − 0.373·14-s + 0.493·15-s − 0.413·16-s − 0.314·17-s − 0.517·18-s − 1.59·19-s − 1.20·20-s − 0.138·21-s − 1.45·22-s − 0.141·23-s + 0.372·24-s − 0.269·25-s + 2.78·26-s − 0.192·27-s + 0.340·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(\approx\) |
\(0.2664307837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2664307837\) |
| \(L(\frac{45}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.04e10T \) |
| good | 2 | \( 1 + 4.60e6T + 8.79e12T^{2} \) |
| 5 | \( 1 + 9.11e14T + 1.13e30T^{2} \) |
| 7 | \( 1 - 3.55e17T + 2.18e36T^{2} \) |
| 11 | \( 1 - 2.29e22T + 6.02e44T^{2} \) |
| 13 | \( 1 + 1.59e24T + 7.93e47T^{2} \) |
| 17 | \( 1 + 8.94e25T + 8.11e52T^{2} \) |
| 19 | \( 1 + 4.96e27T + 9.69e54T^{2} \) |
| 23 | \( 1 + 2.67e28T + 3.58e58T^{2} \) |
| 29 | \( 1 - 5.04e30T + 7.64e62T^{2} \) |
| 31 | \( 1 + 1.04e32T + 1.34e64T^{2} \) |
| 37 | \( 1 - 7.73e33T + 2.70e67T^{2} \) |
| 41 | \( 1 + 8.53e34T + 2.23e69T^{2} \) |
| 43 | \( 1 - 7.83e34T + 1.73e70T^{2} \) |
| 47 | \( 1 - 1.13e36T + 7.94e71T^{2} \) |
| 53 | \( 1 + 1.82e37T + 1.39e74T^{2} \) |
| 59 | \( 1 - 1.43e38T + 1.40e76T^{2} \) |
| 61 | \( 1 + 2.85e38T + 5.87e76T^{2} \) |
| 67 | \( 1 + 3.77e38T + 3.32e78T^{2} \) |
| 71 | \( 1 - 1.14e39T + 4.01e79T^{2} \) |
| 73 | \( 1 - 1.49e40T + 1.32e80T^{2} \) |
| 79 | \( 1 - 7.25e40T + 3.96e81T^{2} \) |
| 83 | \( 1 - 1.45e41T + 3.31e82T^{2} \) |
| 89 | \( 1 - 2.43e41T + 6.66e83T^{2} \) |
| 97 | \( 1 + 4.23e42T + 2.69e85T^{2} \) |
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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
−16.65881590105834763859292760179, −15.02011855028382090769949833700, −12.14776740507623471654883662884, −10.94912573407264739461445184203, −9.492068156160514886933377228658, −7.975597758316548037746401752114, −6.77299151713138569257563704084, −4.42086571027899742197609486222, −1.98538071628631937691766385683, −0.39540860591588220023823076467,
0.39540860591588220023823076467, 1.98538071628631937691766385683, 4.42086571027899742197609486222, 6.77299151713138569257563704084, 7.975597758316548037746401752114, 9.492068156160514886933377228658, 10.94912573407264739461445184203, 12.14776740507623471654883662884, 15.02011855028382090769949833700, 16.65881590105834763859292760179
