L(s) = 1 | + 2-s − 2.61·3-s + 4-s + 0.236·5-s − 2.61·6-s − 7-s + 8-s + 3.85·9-s + 0.236·10-s + 3·11-s − 2.61·12-s − 2.85·13-s − 14-s − 0.618·15-s + 16-s − 4.85·17-s + 3.85·18-s + 0.236·20-s + 2.61·21-s + 3·22-s + 6.70·23-s − 2.61·24-s − 4.94·25-s − 2.85·26-s − 2.23·27-s − 28-s + 3.76·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.51·3-s + 0.5·4-s + 0.105·5-s − 1.06·6-s − 0.377·7-s + 0.353·8-s + 1.28·9-s + 0.0746·10-s + 0.904·11-s − 0.755·12-s − 0.791·13-s − 0.267·14-s − 0.159·15-s + 0.250·16-s − 1.17·17-s + 0.908·18-s + 0.0527·20-s + 0.571·21-s + 0.639·22-s + 1.39·23-s − 0.534·24-s − 0.988·25-s − 0.559·26-s − 0.430·27-s − 0.188·28-s + 0.698·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 - 0.236T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 2.85T + 13T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 7.09T + 41T^{2} \) |
| 43 | \( 1 - 8.56T + 43T^{2} \) |
| 47 | \( 1 + 1.76T + 47T^{2} \) |
| 53 | \( 1 + 0.708T + 53T^{2} \) |
| 59 | \( 1 + 5.76T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 2.09T + 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−7.30327189351335576514860949510, −7.04005691000821066696607345712, −6.12546907394498046253238757389, −5.90055911330653196652715766193, −4.78498716993773470802334171478, −4.55261144070069790623322491264, −3.49457823241203183224752609022, −2.41656594976030202015142720759, −1.26717996519117407440489193144, 0,
1.26717996519117407440489193144, 2.41656594976030202015142720759, 3.49457823241203183224752609022, 4.55261144070069790623322491264, 4.78498716993773470802334171478, 5.90055911330653196652715766193, 6.12546907394498046253238757389, 7.04005691000821066696607345712, 7.30327189351335576514860949510