| L(s) = 1 | + 512·2-s + 2.99e3·3-s + 2.62e5·4-s − 3.82e6·5-s + 1.53e6·6-s − 1.74e8·7-s + 1.34e8·8-s − 1.15e9·9-s − 1.95e9·10-s + 3.32e9·11-s + 7.84e8·12-s − 5.96e10·13-s − 8.93e10·14-s − 1.14e10·15-s + 6.87e10·16-s + 5.44e10·17-s − 5.90e11·18-s − 3.32e11·19-s − 1.00e12·20-s − 5.22e11·21-s + 1.70e12·22-s − 8.66e12·23-s + 4.01e11·24-s − 4.44e12·25-s − 3.05e13·26-s − 6.92e12·27-s − 4.57e13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0877·3-s + 0.5·4-s − 0.875·5-s + 0.0620·6-s − 1.63·7-s + 0.353·8-s − 0.992·9-s − 0.619·10-s + 0.425·11-s + 0.0438·12-s − 1.55·13-s − 1.15·14-s − 0.0768·15-s + 0.250·16-s + 0.111·17-s − 0.701·18-s − 0.236·19-s − 0.437·20-s − 0.143·21-s + 0.301·22-s − 1.00·23-s + 0.0310·24-s − 0.233·25-s − 1.10·26-s − 0.174·27-s − 0.817·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(0.7917650124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7917650124\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 512T \) |
| 29 | \( 1 - 1.45e13T \) |
| good | 3 | \( 1 - 2.99e3T + 1.16e9T^{2} \) |
| 5 | \( 1 + 3.82e6T + 1.90e13T^{2} \) |
| 7 | \( 1 + 1.74e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 3.32e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 5.96e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 5.44e10T + 2.39e23T^{2} \) |
| 19 | \( 1 + 3.32e11T + 1.97e24T^{2} \) |
| 23 | \( 1 + 8.66e12T + 7.46e25T^{2} \) |
| 31 | \( 1 - 3.94e13T + 2.16e28T^{2} \) |
| 37 | \( 1 - 2.66e13T + 6.24e29T^{2} \) |
| 41 | \( 1 + 2.22e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 3.25e15T + 1.08e31T^{2} \) |
| 47 | \( 1 - 6.14e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 2.24e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 5.18e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.80e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 2.15e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 2.92e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 5.96e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 8.08e16T + 1.13e36T^{2} \) |
| 83 | \( 1 - 9.73e17T + 2.90e36T^{2} \) |
| 89 | \( 1 - 2.78e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 3.36e18T + 5.60e37T^{2} \) |
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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
−11.89218794764750673735940885294, −10.34329577539866904099673072765, −9.239267213074654618896002061899, −7.79299057572382480226477558300, −6.71613907235740149973940157336, −5.68916383765707777831933434861, −4.23779327599888274235813219215, −3.28076990545982962489947862412, −2.41201766245914722382911848361, −0.33408420536832338638276445066,
0.33408420536832338638276445066, 2.41201766245914722382911848361, 3.28076990545982962489947862412, 4.23779327599888274235813219215, 5.68916383765707777831933434861, 6.71613907235740149973940157336, 7.79299057572382480226477558300, 9.239267213074654618896002061899, 10.34329577539866904099673072765, 11.89218794764750673735940885294
