| L(s) = 1 | − 0.511·2-s − 2.43·3-s − 1.73·4-s − 2.25·5-s + 1.24·6-s + 1.91·8-s + 2.94·9-s + 1.15·10-s − 11-s + 4.24·12-s − 13-s + 5.49·15-s + 2.50·16-s + 2.82·17-s − 1.50·18-s + 0.755·19-s + 3.91·20-s + 0.511·22-s − 7.61·23-s − 4.66·24-s + 0.0746·25-s + 0.511·26-s + 0.129·27-s − 10.1·29-s − 2.80·30-s − 0.191·31-s − 5.10·32-s + ⋯ |
| L(s) = 1 | − 0.361·2-s − 1.40·3-s − 0.869·4-s − 1.00·5-s + 0.508·6-s + 0.675·8-s + 0.982·9-s + 0.364·10-s − 0.301·11-s + 1.22·12-s − 0.277·13-s + 1.41·15-s + 0.625·16-s + 0.685·17-s − 0.355·18-s + 0.173·19-s + 0.875·20-s + 0.108·22-s − 1.58·23-s − 0.951·24-s + 0.0149·25-s + 0.100·26-s + 0.0249·27-s − 1.87·29-s − 0.512·30-s − 0.0343·31-s − 0.901·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.511T + 2T^{2} \) |
| 3 | \( 1 + 2.43T + 3T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 0.755T + 19T^{2} \) |
| 23 | \( 1 + 7.61T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 0.191T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 1.59T + 43T^{2} \) |
| 47 | \( 1 - 8.33T + 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 - 4.22T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 2.73T + 71T^{2} \) |
| 73 | \( 1 - 6.95T + 73T^{2} \) |
| 79 | \( 1 - 5.84T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 9.18T + 97T^{2} \) |
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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
−7.54097033222318225629967668169, −7.11301486127618103097411148642, −5.96590983000313970394630581550, −5.45833024331347618996542762675, −4.89772785144367158482311064942, −3.96208896297584820856482281064, −3.61352543711610458429495620459, −1.97370868416897756649439213460, −0.71051029677539806204258762728, 0,
0.71051029677539806204258762728, 1.97370868416897756649439213460, 3.61352543711610458429495620459, 3.96208896297584820856482281064, 4.89772785144367158482311064942, 5.45833024331347618996542762675, 5.96590983000313970394630581550, 7.11301486127618103097411148642, 7.54097033222318225629967668169
