| L(s) = 1 | − 2·2-s + 4·4-s − 9.38·5-s − 8·8-s + 18.7·10-s − 20·11-s + 65.6·13-s + 16·16-s − 56.2·17-s + 9.38·19-s − 37.5·20-s + 40·22-s − 48·23-s − 37·25-s − 131.·26-s + 166·29-s − 206.·31-s − 32·32-s + 112.·34-s − 78·37-s − 18.7·38-s + 75.0·40-s − 393.·41-s + 436·43-s − 80·44-s + 96·46-s − 206.·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.839·5-s − 0.353·8-s + 0.593·10-s − 0.548·11-s + 1.40·13-s + 0.250·16-s − 0.803·17-s + 0.113·19-s − 0.419·20-s + 0.387·22-s − 0.435·23-s − 0.295·25-s − 0.990·26-s + 1.06·29-s − 1.19·31-s − 0.176·32-s + 0.567·34-s − 0.346·37-s − 0.0800·38-s + 0.296·40-s − 1.50·41-s + 1.54·43-s − 0.274·44-s + 0.307·46-s − 0.640·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9328433019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9328433019\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 9.38T + 125T^{2} \) |
| 11 | \( 1 + 20T + 1.33e3T^{2} \) |
| 13 | \( 1 - 65.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 56.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 9.38T + 6.85e3T^{2} \) |
| 23 | \( 1 + 48T + 1.21e4T^{2} \) |
| 29 | \( 1 - 166T + 2.43e4T^{2} \) |
| 31 | \( 1 + 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 78T + 5.06e4T^{2} \) |
| 41 | \( 1 + 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 436T + 7.95e4T^{2} \) |
| 47 | \( 1 + 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 62T + 1.48e5T^{2} \) |
| 59 | \( 1 - 666.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 272.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 580T + 3.00e5T^{2} \) |
| 71 | \( 1 - 544T + 3.57e5T^{2} \) |
| 73 | \( 1 + 600.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 680T + 4.93e5T^{2} \) |
| 83 | \( 1 + 196.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 656.T + 9.12e5T^{2} \) |
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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
−9.739796767729301659215511325273, −8.645554856818648397240544249760, −8.286108012076047630829941006302, −7.33497925738591060805815500058, −6.50336696420769706392207862451, −5.49013918813724257453180789283, −4.19094003270597575466085943541, −3.30616827862340583578236505598, −1.95778499536803532488249251807, −0.57944176600634645086332289842,
0.57944176600634645086332289842, 1.95778499536803532488249251807, 3.30616827862340583578236505598, 4.19094003270597575466085943541, 5.49013918813724257453180789283, 6.50336696420769706392207862451, 7.33497925738591060805815500058, 8.286108012076047630829941006302, 8.645554856818648397240544249760, 9.739796767729301659215511325273
