L(s) = 1 | + 2·2-s + 4·4-s + 14·5-s − 7·7-s + 8·8-s + 28·10-s + 28·11-s + 18·13-s − 14·14-s + 16·16-s − 74·17-s + 80·19-s + 56·20-s + 56·22-s + 112·23-s + 71·25-s + 36·26-s − 28·28-s − 190·29-s + 72·31-s + 32·32-s − 148·34-s − 98·35-s − 346·37-s + 160·38-s + 112·40-s − 162·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.25·5-s − 0.377·7-s + 0.353·8-s + 0.885·10-s + 0.767·11-s + 0.384·13-s − 0.267·14-s + 1/4·16-s − 1.05·17-s + 0.965·19-s + 0.626·20-s + 0.542·22-s + 1.01·23-s + 0.567·25-s + 0.271·26-s − 0.188·28-s − 1.21·29-s + 0.417·31-s + 0.176·32-s − 0.746·34-s − 0.473·35-s − 1.53·37-s + 0.683·38-s + 0.442·40-s − 0.617·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.914875756\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.914875756\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
good | 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 18 T + p^{3} T^{2} \) |
| 17 | \( 1 + 74 T + p^{3} T^{2} \) |
| 19 | \( 1 - 80 T + p^{3} T^{2} \) |
| 23 | \( 1 - 112 T + p^{3} T^{2} \) |
| 29 | \( 1 + 190 T + p^{3} T^{2} \) |
| 31 | \( 1 - 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 346 T + p^{3} T^{2} \) |
| 41 | \( 1 + 162 T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 + 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 200 T + p^{3} T^{2} \) |
| 61 | \( 1 + 198 T + p^{3} T^{2} \) |
| 67 | \( 1 + 716 T + p^{3} T^{2} \) |
| 71 | \( 1 + 392 T + p^{3} T^{2} \) |
| 73 | \( 1 - 538 T + p^{3} T^{2} \) |
| 79 | \( 1 - 240 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1072 T + p^{3} T^{2} \) |
| 89 | \( 1 + 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1354 T + p^{3} T^{2} \) |
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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
−13.28579591369027201251467138572, −12.00270826923332995709534680239, −10.93188645914146430158975709895, −9.770377891637989890059260042903, −8.869144705283915979044779159613, −7.01463058785360649024309836730, −6.14339495874195130597288620560, −5.01848454620816188995181327211, −3.36798678013953921936490236760, −1.74512713496138990197795606475,
1.74512713496138990197795606475, 3.36798678013953921936490236760, 5.01848454620816188995181327211, 6.14339495874195130597288620560, 7.01463058785360649024309836730, 8.869144705283915979044779159613, 9.770377891637989890059260042903, 10.93188645914146430158975709895, 12.00270826923332995709534680239, 13.28579591369027201251467138572