| L(s) = 1 | + 3·2-s − 3-s + 4-s − 3·6-s − 7·7-s − 21·8-s − 26·9-s − 9·11-s − 12-s − 13·13-s − 21·14-s − 71·16-s + 6·17-s − 78·18-s − 70·19-s + 7·21-s − 27·22-s − 105·23-s + 21·24-s − 39·26-s + 53·27-s − 7·28-s − 258·29-s + 263·31-s − 45·32-s + 9·33-s + 18·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.192·3-s + 1/8·4-s − 0.204·6-s − 0.377·7-s − 0.928·8-s − 0.962·9-s − 0.246·11-s − 0.0240·12-s − 0.277·13-s − 0.400·14-s − 1.10·16-s + 0.0856·17-s − 1.02·18-s − 0.845·19-s + 0.0727·21-s − 0.261·22-s − 0.951·23-s + 0.178·24-s − 0.294·26-s + 0.377·27-s − 0.0472·28-s − 1.65·29-s + 1.52·31-s − 0.248·32-s + 0.0474·33-s + 0.0907·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.249380124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.249380124\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 13 | \( 1 + p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 3 | \( 1 + T + p^{3} T^{2} \) |
| 11 | \( 1 + 9 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 T + p^{3} T^{2} \) |
| 19 | \( 1 + 70 T + p^{3} T^{2} \) |
| 23 | \( 1 + 105 T + p^{3} T^{2} \) |
| 29 | \( 1 + 258 T + p^{3} T^{2} \) |
| 31 | \( 1 - 263 T + p^{3} T^{2} \) |
| 37 | \( 1 + 371 T + p^{3} T^{2} \) |
| 41 | \( 1 - 111 T + p^{3} T^{2} \) |
| 43 | \( 1 - 538 T + p^{3} T^{2} \) |
| 47 | \( 1 - 411 T + p^{3} T^{2} \) |
| 53 | \( 1 + 60 T + p^{3} T^{2} \) |
| 59 | \( 1 - 156 T + p^{3} T^{2} \) |
| 61 | \( 1 + 529 T + p^{3} T^{2} \) |
| 67 | \( 1 + 173 T + p^{3} T^{2} \) |
| 71 | \( 1 + 372 T + p^{3} T^{2} \) |
| 73 | \( 1 - 727 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1051 T + p^{3} T^{2} \) |
| 83 | \( 1 - 198 T + p^{3} T^{2} \) |
| 89 | \( 1 + 930 T + p^{3} T^{2} \) |
| 97 | \( 1 - 745 T + p^{3} T^{2} \) |
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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
−8.743119133045604377341469212041, −7.86831282611363693597441591308, −6.86472449989485711632534013453, −5.88429502398392091724230533541, −5.69462265335021969693849916684, −4.61648745443959438959673180357, −3.90930299096086872595648619676, −2.99719182034685083102454772565, −2.20267510824453915254523992797, −0.40230898061625131987002489680,
0.40230898061625131987002489680, 2.20267510824453915254523992797, 2.99719182034685083102454772565, 3.90930299096086872595648619676, 4.61648745443959438959673180357, 5.69462265335021969693849916684, 5.88429502398392091724230533541, 6.86472449989485711632534013453, 7.86831282611363693597441591308, 8.743119133045604377341469212041
