| L(s) = 1 | + 0.663·3-s − 1.07·5-s + 2.50·7-s − 2.56·9-s + 2.33·11-s + 3.30·13-s − 0.715·15-s − 4.49·17-s − 1.81·19-s + 1.65·21-s + 1.64·23-s − 3.83·25-s − 3.68·27-s − 7.87·29-s − 4.59·31-s + 1.54·33-s − 2.69·35-s − 5.73·37-s + 2.18·39-s + 10.3·41-s + 9.47·43-s + 2.76·45-s + 10.0·47-s − 0.745·49-s − 2.98·51-s − 11.8·53-s − 2.51·55-s + ⋯ |
| L(s) = 1 | + 0.382·3-s − 0.482·5-s + 0.945·7-s − 0.853·9-s + 0.702·11-s + 0.915·13-s − 0.184·15-s − 1.09·17-s − 0.417·19-s + 0.361·21-s + 0.342·23-s − 0.767·25-s − 0.709·27-s − 1.46·29-s − 0.825·31-s + 0.269·33-s − 0.456·35-s − 0.943·37-s + 0.350·39-s + 1.60·41-s + 1.44·43-s + 0.411·45-s + 1.47·47-s − 0.106·49-s − 0.417·51-s − 1.63·53-s − 0.338·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 - 0.663T + 3T^{2} \) |
| 5 | \( 1 + 1.07T + 5T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 + 4.59T + 31T^{2} \) |
| 37 | \( 1 + 5.73T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 9.47T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 6.38T + 61T^{2} \) |
| 67 | \( 1 - 4.16T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−7.66410113409785572300072114752, −7.32272725396612651124982780473, −6.08668702076734701323599077721, −5.79937570029804658087375688580, −4.61777300444256312304804109759, −4.05643956509567251278042250386, −3.32821007826360770100693632733, −2.25066960918456567176270586871, −1.47107416985235976015658637648, 0,
1.47107416985235976015658637648, 2.25066960918456567176270586871, 3.32821007826360770100693632733, 4.05643956509567251278042250386, 4.61777300444256312304804109759, 5.79937570029804658087375688580, 6.08668702076734701323599077721, 7.32272725396612651124982780473, 7.66410113409785572300072114752
