| L(s) = 1 | + 2-s − 0.639·3-s + 4-s − 1.63·5-s − 0.639·6-s + 1.54·7-s + 8-s − 2.59·9-s − 1.63·10-s − 0.639·12-s + 6.04·13-s + 1.54·14-s + 1.04·15-s + 16-s − 2.54·17-s − 2.59·18-s + 19-s − 1.63·20-s − 0.990·21-s − 7.04·23-s − 0.639·24-s − 2.31·25-s + 6.04·26-s + 3.57·27-s + 1.54·28-s + 1.95·29-s + 1.04·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.369·3-s + 0.5·4-s − 0.733·5-s − 0.260·6-s + 0.585·7-s + 0.353·8-s − 0.863·9-s − 0.518·10-s − 0.184·12-s + 1.67·13-s + 0.414·14-s + 0.270·15-s + 0.250·16-s − 0.618·17-s − 0.610·18-s + 0.229·19-s − 0.366·20-s − 0.216·21-s − 1.46·23-s − 0.130·24-s − 0.462·25-s + 1.18·26-s + 0.687·27-s + 0.292·28-s + 0.362·29-s + 0.191·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.336815440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.336815440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.639T + 3T^{2} \) |
| 5 | \( 1 + 1.63T + 5T^{2} \) |
| 7 | \( 1 - 1.54T + 7T^{2} \) |
| 13 | \( 1 - 6.04T + 13T^{2} \) |
| 17 | \( 1 + 2.54T + 17T^{2} \) |
| 23 | \( 1 + 7.04T + 23T^{2} \) |
| 29 | \( 1 - 1.95T + 29T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 - 5.73T + 37T^{2} \) |
| 41 | \( 1 - 9.13T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 - 3.59T + 47T^{2} \) |
| 53 | \( 1 + 7.95T + 53T^{2} \) |
| 59 | \( 1 - 5.40T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 1.45T + 73T^{2} \) |
| 79 | \( 1 + 1.13T + 79T^{2} \) |
| 83 | \( 1 + 4.36T + 83T^{2} \) |
| 89 | \( 1 + 5.85T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.174997101224785166456933763914, −7.69953472497707007472632000658, −6.66715903923277840474754632546, −5.89461595572480936415458167533, −5.60262149052477330050583802751, −4.30777053859391735256563633832, −4.08004915013758772737223039029, −3.05583947662536031124272865837, −2.05656759438047773569047208272, −0.78044104114830551867073652068,
0.78044104114830551867073652068, 2.05656759438047773569047208272, 3.05583947662536031124272865837, 4.08004915013758772737223039029, 4.30777053859391735256563633832, 5.60262149052477330050583802751, 5.89461595572480936415458167533, 6.66715903923277840474754632546, 7.69953472497707007472632000658, 8.174997101224785166456933763914
