| L(s) = 1 | + 4.77·2-s + 14.7·4-s − 17.3·5-s − 36.1·7-s + 32.3·8-s − 82.8·10-s + 26.7·11-s + 11.9·13-s − 172.·14-s + 36.0·16-s − 90.5·17-s + 19.5·19-s − 256.·20-s + 127.·22-s + 145.·23-s + 176.·25-s + 57.1·26-s − 533.·28-s − 192.·29-s − 69.0·31-s − 86.4·32-s − 432.·34-s + 627.·35-s − 391.·37-s + 93.4·38-s − 561.·40-s + 299.·41-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 1.84·4-s − 1.55·5-s − 1.95·7-s + 1.42·8-s − 2.61·10-s + 0.732·11-s + 0.255·13-s − 3.29·14-s + 0.563·16-s − 1.29·17-s + 0.236·19-s − 2.86·20-s + 1.23·22-s + 1.31·23-s + 1.41·25-s + 0.430·26-s − 3.60·28-s − 1.23·29-s − 0.399·31-s − 0.477·32-s − 2.18·34-s + 3.02·35-s − 1.74·37-s + 0.399·38-s − 2.21·40-s + 1.14·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.634671087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.634671087\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 4.77T + 8T^{2} \) |
| 5 | \( 1 + 17.3T + 125T^{2} \) |
| 7 | \( 1 + 36.1T + 343T^{2} \) |
| 11 | \( 1 - 26.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 90.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 19.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 192.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 69.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 391.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 206.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 581.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 1.78T + 1.48e5T^{2} \) |
| 59 | \( 1 - 34.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 158.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 317.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 90.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 287.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 7.11T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.66e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 539.T + 9.12e5T^{2} \) |
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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
−8.918506650528262970732119852771, −7.38130716223301867179756910242, −6.98766353796050322202358353696, −6.35498388802851418136974189262, −5.49121971385756710185091546349, −4.40002701939842854997102349549, −3.72477177983067082892253346631, −3.41976844001793577066941304747, −2.43093273270171192636497282307, −0.53697362472110949591344861861,
0.53697362472110949591344861861, 2.43093273270171192636497282307, 3.41976844001793577066941304747, 3.72477177983067082892253346631, 4.40002701939842854997102349549, 5.49121971385756710185091546349, 6.35498388802851418136974189262, 6.98766353796050322202358353696, 7.38130716223301867179756910242, 8.918506650528262970732119852771
