| L(s) = 1 | + 2-s − 1.11·3-s + 4-s − 0.774·5-s − 1.11·6-s + 4.10·7-s + 8-s − 1.76·9-s − 0.774·10-s − 11-s − 1.11·12-s + 5.19·13-s + 4.10·14-s + 0.859·15-s + 16-s + 1.38·17-s − 1.76·18-s − 0.774·20-s − 4.55·21-s − 22-s + 3.46·23-s − 1.11·24-s − 4.40·25-s + 5.19·26-s + 5.29·27-s + 4.10·28-s + 2.92·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.640·3-s + 0.5·4-s − 0.346·5-s − 0.453·6-s + 1.55·7-s + 0.353·8-s − 0.589·9-s − 0.244·10-s − 0.301·11-s − 0.320·12-s + 1.44·13-s + 1.09·14-s + 0.221·15-s + 0.250·16-s + 0.335·17-s − 0.416·18-s − 0.173·20-s − 0.994·21-s − 0.213·22-s + 0.722·23-s − 0.226·24-s − 0.880·25-s + 1.01·26-s + 1.01·27-s + 0.775·28-s + 0.542·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.083312859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.083312859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.11T + 3T^{2} \) |
| 5 | \( 1 + 0.774T + 5T^{2} \) |
| 7 | \( 1 - 4.10T + 7T^{2} \) |
| 13 | \( 1 - 5.19T + 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 2.92T + 29T^{2} \) |
| 31 | \( 1 + 3.59T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 - 2.15T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 - 8.22T + 53T^{2} \) |
| 59 | \( 1 - 2.94T + 59T^{2} \) |
| 61 | \( 1 - 3.59T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 5.31T + 71T^{2} \) |
| 73 | \( 1 + 5.61T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 + 9.74T + 83T^{2} \) |
| 89 | \( 1 - 8.13T + 89T^{2} \) |
| 97 | \( 1 + 0.660T + 97T^{2} \) |
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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
−7.73738919961126439433991887768, −7.14050153600991241324795401563, −6.21152233573454937108740733043, −5.56704348184846672510238871103, −5.20294356196177476581412191485, −4.34805643251972704603068322939, −3.72589178678872486898282315928, −2.76813087089245416219699610051, −1.74420522333029767639719340141, −0.849665778910588944123107325378,
0.849665778910588944123107325378, 1.74420522333029767639719340141, 2.76813087089245416219699610051, 3.72589178678872486898282315928, 4.34805643251972704603068322939, 5.20294356196177476581412191485, 5.56704348184846672510238871103, 6.21152233573454937108740733043, 7.14050153600991241324795401563, 7.73738919961126439433991887768
