| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s + 216·6-s + 1.08e3·7-s + 512·8-s + 729·9-s + 6.55e3·11-s + 1.72e3·12-s − 2.52e3·13-s + 8.67e3·14-s + 4.09e3·16-s − 3.04e4·17-s + 5.83e3·18-s + 1.20e4·19-s + 2.92e4·21-s + 5.24e4·22-s + 3.52e3·23-s + 1.38e4·24-s − 2.01e4·26-s + 1.96e4·27-s + 6.93e4·28-s + 8.49e4·29-s − 9.41e4·31-s + 3.27e4·32-s + 1.76e5·33-s − 2.43e5·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.19·7-s + 0.353·8-s + 1/3·9-s + 1.48·11-s + 0.288·12-s − 0.318·13-s + 0.844·14-s + 1/4·16-s − 1.50·17-s + 0.235·18-s + 0.402·19-s + 0.689·21-s + 1.04·22-s + 0.0604·23-s + 0.204·24-s − 0.225·26-s + 0.192·27-s + 0.597·28-s + 0.646·29-s − 0.567·31-s + 0.176·32-s + 0.856·33-s − 1.06·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.069212191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.069212191\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1084 T + p^{7} T^{2} \) |
| 11 | \( 1 - 6552 T + p^{7} T^{2} \) |
| 13 | \( 1 + 194 p T + p^{7} T^{2} \) |
| 17 | \( 1 + 30486 T + p^{7} T^{2} \) |
| 19 | \( 1 - 12020 T + p^{7} T^{2} \) |
| 23 | \( 1 - 3528 T + p^{7} T^{2} \) |
| 29 | \( 1 - 84930 T + p^{7} T^{2} \) |
| 31 | \( 1 + 94168 T + p^{7} T^{2} \) |
| 37 | \( 1 - 509974 T + p^{7} T^{2} \) |
| 41 | \( 1 - 841002 T + p^{7} T^{2} \) |
| 43 | \( 1 + 889052 T + p^{7} T^{2} \) |
| 47 | \( 1 - 852264 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1594482 T + p^{7} T^{2} \) |
| 59 | \( 1 - 752040 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1538702 T + p^{7} T^{2} \) |
| 67 | \( 1 - 947524 T + p^{7} T^{2} \) |
| 71 | \( 1 + 3824928 T + p^{7} T^{2} \) |
| 73 | \( 1 - 913678 T + p^{7} T^{2} \) |
| 79 | \( 1 - 1621880 T + p^{7} T^{2} \) |
| 83 | \( 1 + 1470012 T + p^{7} T^{2} \) |
| 89 | \( 1 - 2375010 T + p^{7} T^{2} \) |
| 97 | \( 1 + 14138786 T + p^{7} T^{2} \) |
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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
−11.64877587118155820979674393439, −11.04866508891095509965825698840, −9.521322712604415836854647006057, −8.536171416653620260790118663672, −7.39560240547660665386222375905, −6.31125192228189490735644930095, −4.77797423129862320076478229963, −3.98982111455905568102244072504, −2.42744504378227882023130517787, −1.28761784308520465735911741877,
1.28761784308520465735911741877, 2.42744504378227882023130517787, 3.98982111455905568102244072504, 4.77797423129862320076478229963, 6.31125192228189490735644930095, 7.39560240547660665386222375905, 8.536171416653620260790118663672, 9.521322712604415836854647006057, 11.04866508891095509965825698840, 11.64877587118155820979674393439
