| L(s) = 1 | − 2·2-s + 5·3-s − 4·4-s + 6·5-s − 10·6-s + 24·8-s − 2·9-s − 12·10-s + 34·11-s − 20·12-s + 57·13-s + 30·15-s − 16·16-s + 80·17-s + 4·18-s + 70·19-s − 24·20-s − 68·22-s + 23·23-s + 120·24-s − 89·25-s − 114·26-s − 145·27-s + 245·29-s − 60·30-s − 103·31-s − 160·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.962·3-s − 1/2·4-s + 0.536·5-s − 0.680·6-s + 1.06·8-s − 0.0740·9-s − 0.379·10-s + 0.931·11-s − 0.481·12-s + 1.21·13-s + 0.516·15-s − 1/4·16-s + 1.14·17-s + 0.0523·18-s + 0.845·19-s − 0.268·20-s − 0.658·22-s + 0.208·23-s + 1.02·24-s − 0.711·25-s − 0.859·26-s − 1.03·27-s + 1.56·29-s − 0.365·30-s − 0.596·31-s − 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.335057213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.335057213\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 23 | \( 1 - p T \) |
| good | 2 | \( 1 + p T + p^{3} T^{2} \) |
| 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 34 T + p^{3} T^{2} \) |
| 13 | \( 1 - 57 T + p^{3} T^{2} \) |
| 17 | \( 1 - 80 T + p^{3} T^{2} \) |
| 19 | \( 1 - 70 T + p^{3} T^{2} \) |
| 29 | \( 1 - 245 T + p^{3} T^{2} \) |
| 31 | \( 1 + 103 T + p^{3} T^{2} \) |
| 37 | \( 1 + 298 T + p^{3} T^{2} \) |
| 41 | \( 1 + 95 T + p^{3} T^{2} \) |
| 43 | \( 1 - 88 T + p^{3} T^{2} \) |
| 47 | \( 1 - 357 T + p^{3} T^{2} \) |
| 53 | \( 1 + 414 T + p^{3} T^{2} \) |
| 59 | \( 1 - 408 T + p^{3} T^{2} \) |
| 61 | \( 1 + 822 T + p^{3} T^{2} \) |
| 67 | \( 1 - 926 T + p^{3} T^{2} \) |
| 71 | \( 1 - 335 T + p^{3} T^{2} \) |
| 73 | \( 1 - 899 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1322 T + p^{3} T^{2} \) |
| 83 | \( 1 - 36 T + p^{3} T^{2} \) |
| 89 | \( 1 - 460 T + p^{3} T^{2} \) |
| 97 | \( 1 - 964 T + p^{3} T^{2} \) |
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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
−9.347265278831962154468950862376, −8.684281858709870916057383361474, −8.145623811305591597253945961623, −7.26630819644795689272943644300, −6.11369006916397024376182009991, −5.21386695427852631264990455798, −3.90220371078706883594373918953, −3.23881356246909613405600607272, −1.77857451426639356592294516615, −0.921343067813029429220734352762,
0.921343067813029429220734352762, 1.77857451426639356592294516615, 3.23881356246909613405600607272, 3.90220371078706883594373918953, 5.21386695427852631264990455798, 6.11369006916397024376182009991, 7.26630819644795689272943644300, 8.145623811305591597253945961623, 8.684281858709870916057383361474, 9.347265278831962154468950862376
