L(s) = 1 | + 2.28·2-s + 1.00·3-s + 3.22·4-s − 5-s + 2.28·6-s + 2.41·7-s + 2.79·8-s − 1.99·9-s − 2.28·10-s + 11-s + 3.22·12-s − 3.24·13-s + 5.52·14-s − 1.00·15-s − 0.0585·16-s + 7.12·17-s − 4.56·18-s + 6.84·19-s − 3.22·20-s + 2.42·21-s + 2.28·22-s + 8.86·23-s + 2.80·24-s + 25-s − 7.41·26-s − 5.00·27-s + 7.78·28-s + ⋯ |
L(s) = 1 | + 1.61·2-s + 0.578·3-s + 1.61·4-s − 0.447·5-s + 0.934·6-s + 0.913·7-s + 0.988·8-s − 0.665·9-s − 0.722·10-s + 0.301·11-s + 0.932·12-s − 0.899·13-s + 1.47·14-s − 0.258·15-s − 0.0146·16-s + 1.72·17-s − 1.07·18-s + 1.57·19-s − 0.720·20-s + 0.528·21-s + 0.487·22-s + 1.84·23-s + 0.571·24-s + 0.200·25-s − 1.45·26-s − 0.963·27-s + 1.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.280963616\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.280963616\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 3 | \( 1 - 1.00T + 3T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 13 | \( 1 + 3.24T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 - 6.84T + 19T^{2} \) |
| 23 | \( 1 - 8.86T + 23T^{2} \) |
| 29 | \( 1 + 8.42T + 29T^{2} \) |
| 31 | \( 1 - 1.03T + 31T^{2} \) |
| 37 | \( 1 - 8.75T + 37T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 - 6.32T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 + 5.01T + 59T^{2} \) |
| 61 | \( 1 - 8.11T + 61T^{2} \) |
| 67 | \( 1 - 4.22T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 6.97T + 89T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−8.104543681643260447170413060482, −7.59800871522841209092425416205, −7.05730159706702259709278225055, −5.79563165009836546771475010539, −5.34620869311425959379512208076, −4.74053864181346388574960922365, −3.78210706907841133687132135108, −3.12730864504839918471037565670, −2.53401783150424981311329410875, −1.19286437204264442176596850674,
1.19286437204264442176596850674, 2.53401783150424981311329410875, 3.12730864504839918471037565670, 3.78210706907841133687132135108, 4.74053864181346388574960922365, 5.34620869311425959379512208076, 5.79563165009836546771475010539, 7.05730159706702259709278225055, 7.59800871522841209092425416205, 8.104543681643260447170413060482