| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s + 103.·5-s − 36·6-s − 64·8-s + 81·9-s − 413.·10-s + 240.·11-s + 144·12-s + 805.·13-s + 931.·15-s + 256·16-s − 1.29e3·17-s − 324·18-s − 275.·19-s + 1.65e3·20-s − 960.·22-s + 3.79e3·23-s − 576·24-s + 7.58e3·25-s − 3.22e3·26-s + 729·27-s + 1.22e3·29-s − 3.72e3·30-s + 5.62e3·31-s − 1.02e3·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.85·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 1.30·10-s + 0.598·11-s + 0.288·12-s + 1.32·13-s + 1.06·15-s + 0.250·16-s − 1.08·17-s − 0.235·18-s − 0.175·19-s + 0.925·20-s − 0.423·22-s + 1.49·23-s − 0.204·24-s + 2.42·25-s − 0.934·26-s + 0.192·27-s + 0.271·29-s − 0.755·30-s + 1.05·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.013829536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.013829536\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 - 9T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 103.T + 3.12e3T^{2} \) |
| 11 | \( 1 - 240.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 805.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.29e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 275.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.79e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.22e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.82e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.17e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.30e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.76e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.60e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.13e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.08e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.96e4T + 8.58e9T^{2} \) |
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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
−10.56175615282824164810600804184, −9.922863915316437720603943394345, −8.811677328464501227164914976979, −8.688139353913898912426923668060, −6.79831465959860249971604943900, −6.35745492810653284218207320620, −5.01240058908124286545879230447, −3.23850774647894959176970162419, −2.01217393699578491740083116183, −1.19157029853969989319827568299,
1.19157029853969989319827568299, 2.01217393699578491740083116183, 3.23850774647894959176970162419, 5.01240058908124286545879230447, 6.35745492810653284218207320620, 6.79831465959860249971604943900, 8.688139353913898912426923668060, 8.811677328464501227164914976979, 9.922863915316437720603943394345, 10.56175615282824164810600804184
