| L(s) = 1 | − 6.78·2-s − 30.1·3-s − 81.9·4-s − 93.2·5-s + 204.·6-s + 1.61e3·7-s + 1.42e3·8-s − 1.27e3·9-s + 632.·10-s + 6.35e3·11-s + 2.47e3·12-s − 1.09e4·14-s + 2.81e3·15-s + 821.·16-s − 8.37e3·17-s + 8.66e3·18-s − 2.12e4·19-s + 7.64e3·20-s − 4.88e4·21-s − 4.31e4·22-s − 1.36e3·23-s − 4.29e4·24-s − 6.94e4·25-s + 1.04e5·27-s − 1.32e5·28-s − 9.65e4·29-s − 1.90e4·30-s + ⋯ |
| L(s) = 1 | − 0.599·2-s − 0.645·3-s − 0.640·4-s − 0.333·5-s + 0.387·6-s + 1.78·7-s + 0.983·8-s − 0.583·9-s + 0.200·10-s + 1.44·11-s + 0.413·12-s − 1.06·14-s + 0.215·15-s + 0.0501·16-s − 0.413·17-s + 0.350·18-s − 0.710·19-s + 0.213·20-s − 1.15·21-s − 0.863·22-s − 0.0234·23-s − 0.634·24-s − 0.888·25-s + 1.02·27-s − 1.14·28-s − 0.735·29-s − 0.129·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.030083387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030083387\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 6.78T + 128T^{2} \) |
| 3 | \( 1 + 30.1T + 2.18e3T^{2} \) |
| 5 | \( 1 + 93.2T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.61e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.35e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 8.37e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.12e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.36e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.65e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.11e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.65e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 9.74e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.03e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.90e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.14e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.81e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.46e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.93e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.12e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.91e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.19e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.73e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.59e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.31e6T + 8.07e13T^{2} \) |
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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
−11.41401298485349894566709401929, −10.66031843778811708298348305341, −9.248444014737087683663733958609, −8.452963925062107279249814217758, −7.64331011809477823449073279423, −6.12323768106926208915482028684, −4.84519286749658382554344335244, −4.09575509280782831358592293218, −1.77287369212817987527398073714, −0.67347878799076636014873043125,
0.67347878799076636014873043125, 1.77287369212817987527398073714, 4.09575509280782831358592293218, 4.84519286749658382554344335244, 6.12323768106926208915482028684, 7.64331011809477823449073279423, 8.452963925062107279249814217758, 9.248444014737087683663733958609, 10.66031843778811708298348305341, 11.41401298485349894566709401929
