| L(s) = 1 | − 8·2-s + 79.2·3-s + 64·4-s + 132.·5-s − 634.·6-s + 1.50e3·7-s − 512·8-s + 4.09e3·9-s − 1.06e3·10-s − 2.92e3·11-s + 5.07e3·12-s − 1.20e4·14-s + 1.05e4·15-s + 4.09e3·16-s − 2.17e4·17-s − 3.27e4·18-s + 5.34e4·19-s + 8.48e3·20-s + 1.19e5·21-s + 2.34e4·22-s − 2.84e4·23-s − 4.05e4·24-s − 6.05e4·25-s + 1.51e5·27-s + 9.65e4·28-s + 1.52e5·29-s − 8.41e4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.69·3-s + 0.5·4-s + 0.474·5-s − 1.19·6-s + 1.66·7-s − 0.353·8-s + 1.87·9-s − 0.335·10-s − 0.662·11-s + 0.847·12-s − 1.17·14-s + 0.804·15-s + 0.250·16-s − 1.07·17-s − 1.32·18-s + 1.78·19-s + 0.237·20-s + 2.81·21-s + 0.468·22-s − 0.487·23-s − 0.599·24-s − 0.774·25-s + 1.48·27-s + 0.831·28-s + 1.16·29-s − 0.568·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.396782450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.396782450\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 79.2T + 2.18e3T^{2} \) |
| 5 | \( 1 - 132.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.50e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.92e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 2.17e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.34e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.84e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.52e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 8.33e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.12e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.10e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.05e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.13e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.89e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.41e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.38e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.69e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.18e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.84e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.07e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.47e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.16e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.23e4T + 8.07e13T^{2} \) |
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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
−10.00420900302895573247433136347, −9.308610118997883025767260406309, −8.266370789128583405676376545125, −8.006791992326456111929166328552, −7.03756303518920195876170155139, −5.37460375742095279129330887643, −4.19496608143432468790937241975, −2.74311444586108485849332207696, −2.05549331540689843672386537320, −1.11169766386835853901130513463,
1.11169766386835853901130513463, 2.05549331540689843672386537320, 2.74311444586108485849332207696, 4.19496608143432468790937241975, 5.37460375742095279129330887643, 7.03756303518920195876170155139, 8.006791992326456111929166328552, 8.266370789128583405676376545125, 9.308610118997883025767260406309, 10.00420900302895573247433136347
