L(s) = 1 | − 1.85·2-s + 3-s + 1.44·4-s + 3.38·5-s − 1.85·6-s − 2.21·7-s + 1.03·8-s + 9-s − 6.28·10-s − 3.17·11-s + 1.44·12-s − 13-s + 4.10·14-s + 3.38·15-s − 4.80·16-s + 2.20·17-s − 1.85·18-s + 0.991·19-s + 4.88·20-s − 2.21·21-s + 5.89·22-s + 8.85·23-s + 1.03·24-s + 6.46·25-s + 1.85·26-s + 27-s − 3.19·28-s + ⋯ |
L(s) = 1 | − 1.31·2-s + 0.577·3-s + 0.721·4-s + 1.51·5-s − 0.757·6-s − 0.836·7-s + 0.365·8-s + 0.333·9-s − 1.98·10-s − 0.957·11-s + 0.416·12-s − 0.277·13-s + 1.09·14-s + 0.874·15-s − 1.20·16-s + 0.533·17-s − 0.437·18-s + 0.227·19-s + 1.09·20-s − 0.483·21-s + 1.25·22-s + 1.84·23-s + 0.210·24-s + 1.29·25-s + 0.363·26-s + 0.192·27-s − 0.603·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.347384158\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.347384158\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 1.85T + 2T^{2} \) |
| 5 | \( 1 - 3.38T + 5T^{2} \) |
| 7 | \( 1 + 2.21T + 7T^{2} \) |
| 11 | \( 1 + 3.17T + 11T^{2} \) |
| 17 | \( 1 - 2.20T + 17T^{2} \) |
| 19 | \( 1 - 0.991T + 19T^{2} \) |
| 23 | \( 1 - 8.85T + 23T^{2} \) |
| 29 | \( 1 + 3.51T + 29T^{2} \) |
| 31 | \( 1 - 3.53T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 + 0.852T + 43T^{2} \) |
| 47 | \( 1 + 7.14T + 47T^{2} \) |
| 53 | \( 1 - 9.36T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 8.02T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 5.39T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 - 2.53T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.702238728349374402732517625788, −7.88478078705255984430324450962, −7.06373302820266299414535608647, −6.60866499877576811932381705988, −5.45565866181352177184225205266, −4.97139916647766948135479670065, −3.46001247879935541772670142236, −2.60993592043429529138303751350, −1.89045461448005928689957075042, −0.794889188344960471016899431760,
0.794889188344960471016899431760, 1.89045461448005928689957075042, 2.60993592043429529138303751350, 3.46001247879935541772670142236, 4.97139916647766948135479670065, 5.45565866181352177184225205266, 6.60866499877576811932381705988, 7.06373302820266299414535608647, 7.88478078705255984430324450962, 8.702238728349374402732517625788