| L(s) = 1 | − 3-s − 3.70·5-s − 1.59·7-s + 9-s − 2.15·11-s − 3.62·13-s + 3.70·15-s + 6.51·17-s − 5.86·19-s + 1.59·21-s − 23-s + 8.74·25-s − 27-s + 29-s + 10.0·31-s + 2.15·33-s + 5.91·35-s + 1.99·37-s + 3.62·39-s − 2.83·41-s − 1.53·43-s − 3.70·45-s − 1.23·47-s − 4.45·49-s − 6.51·51-s − 7.71·53-s + 7.97·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.65·5-s − 0.602·7-s + 0.333·9-s − 0.648·11-s − 1.00·13-s + 0.957·15-s + 1.58·17-s − 1.34·19-s + 0.347·21-s − 0.208·23-s + 1.74·25-s − 0.192·27-s + 0.185·29-s + 1.80·31-s + 0.374·33-s + 0.999·35-s + 0.327·37-s + 0.581·39-s − 0.443·41-s − 0.233·43-s − 0.552·45-s − 0.180·47-s − 0.636·49-s − 0.912·51-s − 1.05·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 + 1.59T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 - 6.51T + 17T^{2} \) |
| 19 | \( 1 + 5.86T + 19T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 1.99T + 37T^{2} \) |
| 41 | \( 1 + 2.83T + 41T^{2} \) |
| 43 | \( 1 + 1.53T + 43T^{2} \) |
| 47 | \( 1 + 1.23T + 47T^{2} \) |
| 53 | \( 1 + 7.71T + 53T^{2} \) |
| 59 | \( 1 - 7.21T + 59T^{2} \) |
| 61 | \( 1 - 1.77T + 61T^{2} \) |
| 67 | \( 1 - 6.75T + 67T^{2} \) |
| 71 | \( 1 - 6.62T + 71T^{2} \) |
| 73 | \( 1 - 9.42T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 9.06T + 89T^{2} \) |
| 97 | \( 1 - 0.456T + 97T^{2} \) |
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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
−7.67943196870374410314239120162, −6.73725455508478692220690071216, −6.31195310712468253177740198689, −5.17248031573011044090756458775, −4.74767361228672635931110269847, −3.89483256368359709158132855319, −3.25833081807587787369817275344, −2.38578916702211903478329458008, −0.828119212387202146088289562628, 0,
0.828119212387202146088289562628, 2.38578916702211903478329458008, 3.25833081807587787369817275344, 3.89483256368359709158132855319, 4.74767361228672635931110269847, 5.17248031573011044090756458775, 6.31195310712468253177740198689, 6.73725455508478692220690071216, 7.67943196870374410314239120162
