| L(s) = 1 | + 36·2-s + 81·3-s + 784·4-s + 2.91e3·6-s + 4.48e3·7-s + 9.79e3·8-s + 6.56e3·9-s + 1.47e3·11-s + 6.35e4·12-s + 1.51e5·13-s + 1.61e5·14-s − 4.88e4·16-s − 1.08e5·17-s + 2.36e5·18-s + 5.93e5·19-s + 3.62e5·21-s + 5.31e4·22-s + 9.69e5·23-s + 7.93e5·24-s + 5.45e6·26-s + 5.31e5·27-s + 3.51e6·28-s − 6.64e6·29-s + 7.07e6·31-s − 6.77e6·32-s + 1.19e5·33-s − 3.89e6·34-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.577·3-s + 1.53·4-s + 0.918·6-s + 0.705·7-s + 0.845·8-s + 1/3·9-s + 0.0303·11-s + 0.884·12-s + 1.47·13-s + 1.12·14-s − 0.186·16-s − 0.314·17-s + 0.530·18-s + 1.04·19-s + 0.407·21-s + 0.0483·22-s + 0.722·23-s + 0.487·24-s + 2.34·26-s + 0.192·27-s + 1.07·28-s − 1.74·29-s + 1.37·31-s − 1.14·32-s + 0.0175·33-s − 0.499·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(7.073560860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.073560860\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{4} T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 9 p^{2} T + p^{9} T^{2} \) |
| 7 | \( 1 - 640 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 1476 T + p^{9} T^{2} \) |
| 13 | \( 1 - 151522 T + p^{9} T^{2} \) |
| 17 | \( 1 + 108162 T + p^{9} T^{2} \) |
| 19 | \( 1 - 593084 T + p^{9} T^{2} \) |
| 23 | \( 1 - 969480 T + p^{9} T^{2} \) |
| 29 | \( 1 + 6642522 T + p^{9} T^{2} \) |
| 31 | \( 1 - 7070600 T + p^{9} T^{2} \) |
| 37 | \( 1 - 7472410 T + p^{9} T^{2} \) |
| 41 | \( 1 + 4350150 T + p^{9} T^{2} \) |
| 43 | \( 1 - 4358716 T + p^{9} T^{2} \) |
| 47 | \( 1 + 28309248 T + p^{9} T^{2} \) |
| 53 | \( 1 + 16111710 T + p^{9} T^{2} \) |
| 59 | \( 1 + 86075964 T + p^{9} T^{2} \) |
| 61 | \( 1 - 32213918 T + p^{9} T^{2} \) |
| 67 | \( 1 + 99531452 T + p^{9} T^{2} \) |
| 71 | \( 1 + 44170488 T + p^{9} T^{2} \) |
| 73 | \( 1 - 23560630 T + p^{9} T^{2} \) |
| 79 | \( 1 + 401754760 T + p^{9} T^{2} \) |
| 83 | \( 1 - 744528708 T + p^{9} T^{2} \) |
| 89 | \( 1 - 769871034 T + p^{9} T^{2} \) |
| 97 | \( 1 + 907130882 T + p^{9} T^{2} \) |
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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
−13.10515263175056403946876123468, −11.74055885475959607910462963144, −10.96221548863570279189002996048, −9.181160458616554358355140557508, −7.85925942453400161148532561548, −6.45399998333194366000836279605, −5.22180577226831312563194813947, −4.04557260142006359407968615629, −2.99058161146915963692286323639, −1.48847922712525936824374035083,
1.48847922712525936824374035083, 2.99058161146915963692286323639, 4.04557260142006359407968615629, 5.22180577226831312563194813947, 6.45399998333194366000836279605, 7.85925942453400161148532561548, 9.181160458616554358355140557508, 10.96221548863570279189002996048, 11.74055885475959607910462963144, 13.10515263175056403946876123468
