| L(s) = 1 | − 1.28·2-s − 79.7·3-s − 126.·4-s + 102.·6-s + 538.·7-s + 326.·8-s + 4.17e3·9-s − 1.21e3·11-s + 1.00e4·12-s − 7.07e3·13-s − 690.·14-s + 1.57e4·16-s + 3.34e3·17-s − 5.34e3·18-s + 2.21e4·19-s − 4.29e4·21-s + 1.55e3·22-s + 5.85e4·23-s − 2.60e4·24-s + 9.06e3·26-s − 1.58e5·27-s − 6.80e4·28-s − 2.06e5·29-s + 1.77e5·31-s − 6.19e4·32-s + 9.68e4·33-s − 4.29e3·34-s + ⋯ |
| L(s) = 1 | − 0.113·2-s − 1.70·3-s − 0.987·4-s + 0.193·6-s + 0.593·7-s + 0.225·8-s + 1.90·9-s − 0.275·11-s + 1.68·12-s − 0.892·13-s − 0.0672·14-s + 0.961·16-s + 0.165·17-s − 0.216·18-s + 0.741·19-s − 1.01·21-s + 0.0311·22-s + 1.00·23-s − 0.384·24-s + 0.101·26-s − 1.54·27-s − 0.585·28-s − 1.57·29-s + 1.07·31-s − 0.334·32-s + 0.469·33-s − 0.0187·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6255327452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6255327452\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 1.28T + 128T^{2} \) |
| 3 | \( 1 + 79.7T + 2.18e3T^{2} \) |
| 7 | \( 1 - 538.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.21e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.07e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.34e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.21e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.85e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.06e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.77e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.84e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.27e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.64e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.49e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.30e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.42e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.66e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.95e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 9.21e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.28e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.17e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.42e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.59e6T + 8.07e13T^{2} \) |
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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
−16.45570389858501316292858202201, −14.80743303208370292771912600155, −13.18593931280487769310645509357, −12.03482885294758751663402442479, −10.84506926065475156180025010945, −9.556243959847314121512723589077, −7.52756684251553284047092052593, −5.59498244385591433715280723129, −4.61461303766045717788779287552, −0.76452271497330909346391706013,
0.76452271497330909346391706013, 4.61461303766045717788779287552, 5.59498244385591433715280723129, 7.52756684251553284047092052593, 9.556243959847314121512723589077, 10.84506926065475156180025010945, 12.03482885294758751663402442479, 13.18593931280487769310645509357, 14.80743303208370292771912600155, 16.45570389858501316292858202201
