| L(s) = 1 | − 9·3-s + 6·5-s + 176·7-s + 81·9-s + 496·11-s − 178·13-s − 54·15-s + 202·17-s + 361·19-s − 1.58e3·21-s − 4.39e3·23-s − 3.08e3·25-s − 729·27-s − 5.90e3·29-s − 5.76e3·31-s − 4.46e3·33-s + 1.05e3·35-s − 3.90e3·37-s + 1.60e3·39-s + 1.57e4·41-s + 7.49e3·43-s + 486·45-s + 7.45e3·47-s + 1.41e4·49-s − 1.81e3·51-s − 2.90e4·53-s + 2.97e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.107·5-s + 1.35·7-s + 1/3·9-s + 1.23·11-s − 0.292·13-s − 0.0619·15-s + 0.169·17-s + 0.229·19-s − 0.783·21-s − 1.73·23-s − 0.988·25-s − 0.192·27-s − 1.30·29-s − 1.07·31-s − 0.713·33-s + 0.145·35-s − 0.469·37-s + 0.168·39-s + 1.46·41-s + 0.617·43-s + 0.0357·45-s + 0.492·47-s + 0.843·49-s − 0.0978·51-s − 1.41·53-s + 0.132·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 19 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 6 T + p^{5} T^{2} \) |
| 7 | \( 1 - 176 T + p^{5} T^{2} \) |
| 11 | \( 1 - 496 T + p^{5} T^{2} \) |
| 13 | \( 1 + 178 T + p^{5} T^{2} \) |
| 17 | \( 1 - 202 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4396 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5902 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5760 T + p^{5} T^{2} \) |
| 37 | \( 1 + 3906 T + p^{5} T^{2} \) |
| 41 | \( 1 - 15774 T + p^{5} T^{2} \) |
| 43 | \( 1 - 7492 T + p^{5} T^{2} \) |
| 47 | \( 1 - 7452 T + p^{5} T^{2} \) |
| 53 | \( 1 + 29014 T + p^{5} T^{2} \) |
| 59 | \( 1 + 13604 T + p^{5} T^{2} \) |
| 61 | \( 1 + 12466 T + p^{5} T^{2} \) |
| 67 | \( 1 + 43436 T + p^{5} T^{2} \) |
| 71 | \( 1 + 28800 T + p^{5} T^{2} \) |
| 73 | \( 1 - 80746 T + p^{5} T^{2} \) |
| 79 | \( 1 + 76456 T + p^{5} T^{2} \) |
| 83 | \( 1 - 56880 T + p^{5} T^{2} \) |
| 89 | \( 1 + 103266 T + p^{5} T^{2} \) |
| 97 | \( 1 - 82490 T + p^{5} T^{2} \) |
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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
−9.056281555896549362488557780461, −7.88797669840042100627490203610, −7.39609766564309295077634227903, −6.14172996889258848520354692323, −5.54751281675704155194461955199, −4.45582153386018497022935792801, −3.78833221534871123661170422396, −2.02411252949137011883488869031, −1.38057309042500328375904190432, 0,
1.38057309042500328375904190432, 2.02411252949137011883488869031, 3.78833221534871123661170422396, 4.45582153386018497022935792801, 5.54751281675704155194461955199, 6.14172996889258848520354692323, 7.39609766564309295077634227903, 7.88797669840042100627490203610, 9.056281555896549362488557780461
