| L(s) = 1 | + 1.02e18·2-s − 1.22e29·3-s − 1.60e36·4-s − 2.11e42·5-s − 1.25e47·6-s − 1.42e51·7-s − 4.38e54·8-s + 9.58e57·9-s − 2.17e60·10-s − 1.31e63·11-s + 1.95e65·12-s + 4.76e67·13-s − 1.46e69·14-s + 2.58e71·15-s − 2.50e71·16-s + 1.88e74·17-s + 9.86e75·18-s − 6.84e76·19-s + 3.37e78·20-s + 1.73e80·21-s − 1.35e81·22-s − 2.52e82·23-s + 5.36e83·24-s + 6.90e83·25-s + 4.89e85·26-s − 5.13e86·27-s + 2.27e87·28-s + ⋯ |
| L(s) = 1 | + 0.630·2-s − 1.66·3-s − 0.602·4-s − 1.08·5-s − 1.05·6-s − 1.05·7-s − 1.01·8-s + 1.77·9-s − 0.686·10-s − 1.30·11-s + 1.00·12-s + 1.92·13-s − 0.666·14-s + 1.81·15-s − 0.0355·16-s + 0.682·17-s + 1.12·18-s − 0.295·19-s + 0.654·20-s + 1.76·21-s − 0.823·22-s − 1.04·23-s + 1.68·24-s + 0.183·25-s + 1.21·26-s − 1.29·27-s + 0.636·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(122-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+60.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(61)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{123}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.02e18T + 2.65e36T^{2} \) |
| 3 | \( 1 + 1.22e29T + 5.39e57T^{2} \) |
| 5 | \( 1 + 2.11e42T + 3.76e84T^{2} \) |
| 7 | \( 1 + 1.42e51T + 1.80e102T^{2} \) |
| 11 | \( 1 + 1.31e63T + 1.01e126T^{2} \) |
| 13 | \( 1 - 4.76e67T + 6.12e134T^{2} \) |
| 17 | \( 1 - 1.88e74T + 7.66e148T^{2} \) |
| 19 | \( 1 + 6.84e76T + 5.36e154T^{2} \) |
| 23 | \( 1 + 2.52e82T + 5.87e164T^{2} \) |
| 29 | \( 1 + 9.19e87T + 8.91e176T^{2} \) |
| 31 | \( 1 - 1.76e90T + 2.84e180T^{2} \) |
| 37 | \( 1 - 1.94e94T + 5.65e189T^{2} \) |
| 41 | \( 1 - 4.01e97T + 1.40e195T^{2} \) |
| 43 | \( 1 + 5.59e98T + 4.46e197T^{2} \) |
| 47 | \( 1 + 6.96e100T + 2.10e202T^{2} \) |
| 53 | \( 1 - 1.28e102T + 4.33e208T^{2} \) |
| 59 | \( 1 + 9.94e106T + 1.87e214T^{2} \) |
| 61 | \( 1 - 1.94e108T + 1.05e216T^{2} \) |
| 67 | \( 1 - 5.84e109T + 9.01e220T^{2} \) |
| 71 | \( 1 + 1.26e112T + 1.00e224T^{2} \) |
| 73 | \( 1 - 5.17e112T + 2.89e225T^{2} \) |
| 79 | \( 1 - 4.13e114T + 4.10e229T^{2} \) |
| 83 | \( 1 + 7.91e115T + 1.61e232T^{2} \) |
| 89 | \( 1 - 1.78e117T + 7.51e235T^{2} \) |
| 97 | \( 1 + 6.06e119T + 2.50e240T^{2} \) |
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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
−12.45788454362368449136162504133, −11.34926180087994570873111127915, −10.09488983570226105522333939606, −8.128134937716624180771925477415, −6.34370635410831327215176067722, −5.58153379699633283599709475076, −4.29085246711873857986566204994, −3.37168855217321009935334229082, −0.74371437155340444799141446063, 0,
0.74371437155340444799141446063, 3.37168855217321009935334229082, 4.29085246711873857986566204994, 5.58153379699633283599709475076, 6.34370635410831327215176067722, 8.128134937716624180771925477415, 10.09488983570226105522333939606, 11.34926180087994570873111127915, 12.45788454362368449136162504133
