| L(s) = 1 | − 2-s + 1.38·3-s + 4-s + 3.52·5-s − 1.38·6-s − 0.0942·7-s − 8-s − 1.08·9-s − 3.52·10-s − 4.45·11-s + 1.38·12-s + 0.0942·14-s + 4.87·15-s + 16-s − 0.332·17-s + 1.08·18-s + 19-s + 3.52·20-s − 0.130·21-s + 4.45·22-s + 2.61·23-s − 1.38·24-s + 7.43·25-s − 5.65·27-s − 0.0942·28-s − 10.0·29-s − 4.87·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.798·3-s + 0.5·4-s + 1.57·5-s − 0.564·6-s − 0.0356·7-s − 0.353·8-s − 0.362·9-s − 1.11·10-s − 1.34·11-s + 0.399·12-s + 0.0252·14-s + 1.25·15-s + 0.250·16-s − 0.0805·17-s + 0.256·18-s + 0.229·19-s + 0.788·20-s − 0.0284·21-s + 0.949·22-s + 0.546·23-s − 0.282·24-s + 1.48·25-s − 1.08·27-s − 0.0178·28-s − 1.86·29-s − 0.890·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.38T + 3T^{2} \) |
| 5 | \( 1 - 3.52T + 5T^{2} \) |
| 7 | \( 1 + 0.0942T + 7T^{2} \) |
| 11 | \( 1 + 4.45T + 11T^{2} \) |
| 17 | \( 1 + 0.332T + 17T^{2} \) |
| 23 | \( 1 - 2.61T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 8.91T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 - 7.56T + 43T^{2} \) |
| 47 | \( 1 - 8.14T + 47T^{2} \) |
| 53 | \( 1 + 1.54T + 53T^{2} \) |
| 59 | \( 1 + 6.56T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 + 6.61T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 6.32T + 73T^{2} \) |
| 79 | \( 1 - 0.875T + 79T^{2} \) |
| 83 | \( 1 - 0.100T + 83T^{2} \) |
| 89 | \( 1 + 1.93T + 89T^{2} \) |
| 97 | \( 1 + 4.94T + 97T^{2} \) |
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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
−7.55376955953591558105330422330, −7.35616631596548438354807065214, −6.20707117389699899727111476778, −5.53491848011407747781752236823, −5.19871450062171266750309568632, −3.67925035485736783310627282844, −2.85872998787244452384263081474, −2.21686004603988544603778759753, −1.61595241049805349074835655462, 0,
1.61595241049805349074835655462, 2.21686004603988544603778759753, 2.85872998787244452384263081474, 3.67925035485736783310627282844, 5.19871450062171266750309568632, 5.53491848011407747781752236823, 6.20707117389699899727111476778, 7.35616631596548438354807065214, 7.55376955953591558105330422330
