| L(s) = 1 | + 91.8·3-s + 432.·5-s − 1.48e3·7-s + 6.25e3·9-s + 5.78e3·11-s − 2.54e3·13-s + 3.97e4·15-s − 5.28e3·17-s + 6.85e3·19-s − 1.36e5·21-s − 2.71e4·23-s + 1.08e5·25-s + 3.74e5·27-s − 2.40e5·29-s + 1.74e4·31-s + 5.31e5·33-s − 6.41e5·35-s + 2.26e4·37-s − 2.33e5·39-s + 5.98e4·41-s + 9.30e4·43-s + 2.70e6·45-s + 8.18e5·47-s + 1.37e6·49-s − 4.85e5·51-s − 1.28e6·53-s + 2.50e6·55-s + ⋯ |
| L(s) = 1 | + 1.96·3-s + 1.54·5-s − 1.63·7-s + 2.86·9-s + 1.31·11-s − 0.321·13-s + 3.03·15-s − 0.260·17-s + 0.229·19-s − 3.21·21-s − 0.465·23-s + 1.39·25-s + 3.65·27-s − 1.83·29-s + 0.105·31-s + 2.57·33-s − 2.52·35-s + 0.0735·37-s − 0.630·39-s + 0.135·41-s + 0.178·43-s + 4.42·45-s + 1.14·47-s + 1.66·49-s − 0.512·51-s − 1.18·53-s + 2.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.510951961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.510951961\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 6.85e3T \) |
| good | 3 | \( 1 - 91.8T + 2.18e3T^{2} \) |
| 5 | \( 1 - 432.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.48e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.78e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.54e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 5.28e3T + 4.10e8T^{2} \) |
| 23 | \( 1 + 2.71e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.40e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.74e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.26e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.98e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 9.30e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.18e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.28e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.17e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 6.32e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.49e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 6.09e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.72e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.24e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.14e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.02e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.02e6T + 8.07e13T^{2} \) |
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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
−13.33799990391367772269873552088, −12.60524913179758638183851826518, −10.12060833325098516811020255338, −9.413289509533806117735027669398, −9.039213842306596096636640734546, −7.22763311952955893015052643402, −6.18657122260240903635275601144, −3.88111449807515538802321559004, −2.75397476722405218750834688768, −1.63821292606797216676389609596,
1.63821292606797216676389609596, 2.75397476722405218750834688768, 3.88111449807515538802321559004, 6.18657122260240903635275601144, 7.22763311952955893015052643402, 9.039213842306596096636640734546, 9.413289509533806117735027669398, 10.12060833325098516811020255338, 12.60524913179758638183851826518, 13.33799990391367772269873552088
