L(s) = 1 | − 2.53e8·2-s − 1.66e12·3-s + 2.82e16·4-s + 1.07e19·5-s + 4.23e20·6-s − 1.10e23·7-s + 1.97e24·8-s − 1.71e26·9-s − 2.71e27·10-s − 7.94e28·11-s − 4.71e28·12-s + 3.15e30·13-s + 2.79e31·14-s − 1.79e31·15-s − 1.51e33·16-s + 3.96e33·17-s + 4.35e34·18-s + 1.06e35·19-s + 3.02e35·20-s + 1.84e35·21-s + 2.01e37·22-s + 2.36e37·23-s − 3.29e36·24-s − 1.62e38·25-s − 8.00e38·26-s + 5.77e38·27-s − 3.11e39·28-s + ⋯ |
L(s) = 1 | − 1.33·2-s − 0.126·3-s + 0.783·4-s + 0.643·5-s + 0.168·6-s − 0.634·7-s + 0.288·8-s − 0.984·9-s − 0.859·10-s − 1.82·11-s − 0.0990·12-s + 0.734·13-s + 0.847·14-s − 0.0813·15-s − 1.16·16-s + 0.576·17-s + 1.31·18-s + 0.729·19-s + 0.504·20-s + 0.0802·21-s + 2.43·22-s + 0.844·23-s − 0.0364·24-s − 0.585·25-s − 0.981·26-s + 0.250·27-s − 0.497·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(56-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+55/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(28)\) |
\(\approx\) |
\(0.6574037555\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6574037555\) |
\(L(\frac{57}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
good | 2 | \( 1 + 2.53e8T + 3.60e16T^{2} \) |
| 3 | \( 1 + 1.66e12T + 1.74e26T^{2} \) |
| 5 | \( 1 - 1.07e19T + 2.77e38T^{2} \) |
| 7 | \( 1 + 1.10e23T + 3.02e46T^{2} \) |
| 11 | \( 1 + 7.94e28T + 1.89e57T^{2} \) |
| 13 | \( 1 - 3.15e30T + 1.84e61T^{2} \) |
| 17 | \( 1 - 3.96e33T + 4.72e67T^{2} \) |
| 19 | \( 1 - 1.06e35T + 2.14e70T^{2} \) |
| 23 | \( 1 - 2.36e37T + 7.85e74T^{2} \) |
| 29 | \( 1 + 2.07e40T + 2.70e80T^{2} \) |
| 31 | \( 1 - 1.26e41T + 1.05e82T^{2} \) |
| 37 | \( 1 - 1.05e43T + 1.78e86T^{2} \) |
| 41 | \( 1 - 3.96e44T + 5.04e88T^{2} \) |
| 43 | \( 1 - 4.12e44T + 6.93e89T^{2} \) |
| 47 | \( 1 - 1.78e46T + 9.23e91T^{2} \) |
| 53 | \( 1 + 1.03e47T + 6.84e94T^{2} \) |
| 59 | \( 1 - 3.49e48T + 2.49e97T^{2} \) |
| 61 | \( 1 + 1.91e48T + 1.56e98T^{2} \) |
| 67 | \( 1 - 1.84e49T + 2.71e100T^{2} \) |
| 71 | \( 1 - 7.05e50T + 6.59e101T^{2} \) |
| 73 | \( 1 + 2.58e51T + 3.03e102T^{2} \) |
| 79 | \( 1 - 9.12e51T + 2.34e104T^{2} \) |
| 83 | \( 1 - 4.34e52T + 3.54e105T^{2} \) |
| 89 | \( 1 - 4.67e53T + 1.64e107T^{2} \) |
| 97 | \( 1 + 8.39e53T + 1.87e109T^{2} \) |
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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
−18.86303680419148520095675615549, −17.59401341441945158428791142920, −16.10315799827423832519098423060, −13.41925023132952489917618904491, −10.81170957746528457970535173058, −9.438438160703926861457700939837, −7.84000730608031809753268719439, −5.68326504619685570106718169026, −2.62468760178042997638962466692, −0.67796267284066736667574685265,
0.67796267284066736667574685265, 2.62468760178042997638962466692, 5.68326504619685570106718169026, 7.84000730608031809753268719439, 9.438438160703926861457700939837, 10.81170957746528457970535173058, 13.41925023132952489917618904491, 16.10315799827423832519098423060, 17.59401341441945158428791142920, 18.86303680419148520095675615549