| L(s) = 1 | + 2-s − 0.303·3-s + 4-s − 0.303·6-s − 1.08·7-s + 8-s − 2.90·9-s + 0.403·11-s − 0.303·12-s − 1.08·14-s + 16-s − 5.21·17-s − 2.90·18-s − 0.844·19-s + 0.329·21-s + 0.403·22-s − 1.76·23-s − 0.303·24-s + 1.79·27-s − 1.08·28-s − 5.96·29-s + 5.25·31-s + 32-s − 0.122·33-s − 5.21·34-s − 2.90·36-s − 4.42·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.175·3-s + 0.5·4-s − 0.123·6-s − 0.410·7-s + 0.353·8-s − 0.969·9-s + 0.121·11-s − 0.0876·12-s − 0.290·14-s + 0.250·16-s − 1.26·17-s − 0.685·18-s − 0.193·19-s + 0.0719·21-s + 0.0861·22-s − 0.367·23-s − 0.0619·24-s + 0.345·27-s − 0.205·28-s − 1.10·29-s + 0.943·31-s + 0.176·32-s − 0.0213·33-s − 0.894·34-s − 0.484·36-s − 0.727·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.138564679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.138564679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.303T + 3T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 11 | \( 1 - 0.403T + 11T^{2} \) |
| 17 | \( 1 + 5.21T + 17T^{2} \) |
| 19 | \( 1 + 0.844T + 19T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 + 5.96T + 29T^{2} \) |
| 31 | \( 1 - 5.25T + 31T^{2} \) |
| 37 | \( 1 + 4.42T + 37T^{2} \) |
| 41 | \( 1 + 3.20T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 7.28T + 59T^{2} \) |
| 61 | \( 1 + 2.64T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 2.92T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 6.99T + 89T^{2} \) |
| 97 | \( 1 - 9.47T + 97T^{2} \) |
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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
−7.65734391222985073772159552447, −6.88519194544538966096418154033, −6.33817104173123931743357765148, −5.69707700431443733085603557299, −5.09489712180311646794445220009, −4.17713795435445212031953889917, −3.63549365449262839049818514620, −2.62944444765912612538265682673, −2.10774627931413773479788073596, −0.61576753353956929855623545361,
0.61576753353956929855623545361, 2.10774627931413773479788073596, 2.62944444765912612538265682673, 3.63549365449262839049818514620, 4.17713795435445212031953889917, 5.09489712180311646794445220009, 5.69707700431443733085603557299, 6.33817104173123931743357765148, 6.88519194544538966096418154033, 7.65734391222985073772159552447
