L(s) = 1 | − 1.85·2-s + 1.40·3-s + 1.43·4-s − 4.08·5-s − 2.60·6-s + 1.71·7-s + 1.04·8-s − 1.02·9-s + 7.57·10-s + 5.46·11-s + 2.01·12-s − 5.16·13-s − 3.17·14-s − 5.74·15-s − 4.81·16-s − 17-s + 1.89·18-s + 6.66·19-s − 5.86·20-s + 2.40·21-s − 10.1·22-s − 6.04·23-s + 1.47·24-s + 11.7·25-s + 9.57·26-s − 5.65·27-s + 2.45·28-s + ⋯ |
L(s) = 1 | − 1.31·2-s + 0.811·3-s + 0.717·4-s − 1.82·5-s − 1.06·6-s + 0.647·7-s + 0.370·8-s − 0.340·9-s + 2.39·10-s + 1.64·11-s + 0.582·12-s − 1.43·13-s − 0.848·14-s − 1.48·15-s − 1.20·16-s − 0.242·17-s + 0.446·18-s + 1.52·19-s − 1.31·20-s + 0.525·21-s − 2.15·22-s − 1.26·23-s + 0.300·24-s + 2.34·25-s + 1.87·26-s − 1.08·27-s + 0.464·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 2 | \( 1 + 1.85T + 2T^{2} \) |
| 3 | \( 1 - 1.40T + 3T^{2} \) |
| 5 | \( 1 + 4.08T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 + 5.16T + 13T^{2} \) |
| 19 | \( 1 - 6.66T + 19T^{2} \) |
| 23 | \( 1 + 6.04T + 23T^{2} \) |
| 29 | \( 1 + 1.29T + 29T^{2} \) |
| 31 | \( 1 - 7.77T + 31T^{2} \) |
| 37 | \( 1 + 2.94T + 37T^{2} \) |
| 41 | \( 1 + 8.37T + 41T^{2} \) |
| 43 | \( 1 + 4.55T + 43T^{2} \) |
| 47 | \( 1 + 3.75T + 47T^{2} \) |
| 53 | \( 1 + 9.86T + 53T^{2} \) |
| 61 | \( 1 - 3.09T + 61T^{2} \) |
| 67 | \( 1 - 6.99T + 67T^{2} \) |
| 71 | \( 1 + 9.77T + 71T^{2} \) |
| 73 | \( 1 + 8.11T + 73T^{2} \) |
| 79 | \( 1 + 3.50T + 79T^{2} \) |
| 83 | \( 1 + 6.36T + 83T^{2} \) |
| 89 | \( 1 + 5.97T + 89T^{2} \) |
| 97 | \( 1 - 6.81T + 97T^{2} \) |
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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
−9.417464883189236214680905997807, −8.504251107269394191245160048468, −8.126536725006030564753987382815, −7.47912470424065892179383114254, −6.78247662427744509695873702639, −4.88446824946440935317693635711, −4.06490121626915188588328194355, −3.05874353313088261646516539285, −1.53627026630342604577470159166, 0,
1.53627026630342604577470159166, 3.05874353313088261646516539285, 4.06490121626915188588328194355, 4.88446824946440935317693635711, 6.78247662427744509695873702639, 7.47912470424065892179383114254, 8.126536725006030564753987382815, 8.504251107269394191245160048468, 9.417464883189236214680905997807