| L(s) = 1 | + 2-s − 2.90·3-s + 4-s − 4.15·5-s − 2.90·6-s + 3.02·7-s + 8-s + 5.46·9-s − 4.15·10-s + 1.94·11-s − 2.90·12-s + 7.15·13-s + 3.02·14-s + 12.0·15-s + 16-s + 1.01·17-s + 5.46·18-s − 19-s − 4.15·20-s − 8.78·21-s + 1.94·22-s + 3.81·23-s − 2.90·24-s + 12.2·25-s + 7.15·26-s − 7.17·27-s + 3.02·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.67·3-s + 0.5·4-s − 1.85·5-s − 1.18·6-s + 1.14·7-s + 0.353·8-s + 1.82·9-s − 1.31·10-s + 0.587·11-s − 0.839·12-s + 1.98·13-s + 0.807·14-s + 3.11·15-s + 0.250·16-s + 0.247·17-s + 1.28·18-s − 0.229·19-s − 0.928·20-s − 1.91·21-s + 0.415·22-s + 0.795·23-s − 0.593·24-s + 2.44·25-s + 1.40·26-s − 1.38·27-s + 0.570·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.030319901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.030319901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 5 | \( 1 + 4.15T + 5T^{2} \) |
| 7 | \( 1 - 3.02T + 7T^{2} \) |
| 11 | \( 1 - 1.94T + 11T^{2} \) |
| 13 | \( 1 - 7.15T + 13T^{2} \) |
| 17 | \( 1 - 1.01T + 17T^{2} \) |
| 23 | \( 1 - 3.81T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 - 9.42T + 37T^{2} \) |
| 41 | \( 1 - 0.697T + 41T^{2} \) |
| 43 | \( 1 - 7.53T + 43T^{2} \) |
| 47 | \( 1 + 5.51T + 47T^{2} \) |
| 53 | \( 1 + 3.63T + 53T^{2} \) |
| 59 | \( 1 - 7.99T + 59T^{2} \) |
| 61 | \( 1 + 1.55T + 61T^{2} \) |
| 67 | \( 1 - 1.68T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 3.13T + 73T^{2} \) |
| 79 | \( 1 + 1.68T + 79T^{2} \) |
| 83 | \( 1 + 5.75T + 83T^{2} \) |
| 89 | \( 1 + 8.16T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.77878742197896617820813925929, −6.85700255390219704438537525232, −6.42134693480717038852386822862, −5.73446191070808294004061083080, −4.79688584682475912303670301291, −4.44371303078621444850247155385, −3.93410420732722897855609462121, −3.00629448038752147711813438463, −1.21851025714975922792933211902, −0.881104887502317971773292226972,
0.881104887502317971773292226972, 1.21851025714975922792933211902, 3.00629448038752147711813438463, 3.93410420732722897855609462121, 4.44371303078621444850247155385, 4.79688584682475912303670301291, 5.73446191070808294004061083080, 6.42134693480717038852386822862, 6.85700255390219704438537525232, 7.77878742197896617820813925929
