| L(s) = 1 | − 2-s + 4-s − 8-s − 3·11-s + 16-s + 17-s − 2·19-s + 3·22-s − 9·23-s − 29-s − 10·31-s − 32-s − 34-s + 8·37-s + 2·38-s + 4·41-s + 8·43-s − 3·44-s + 9·46-s + 10·47-s − 7·49-s + 58-s − 10·59-s + 7·61-s + 10·62-s + 64-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.904·11-s + 1/4·16-s + 0.242·17-s − 0.458·19-s + 0.639·22-s − 1.87·23-s − 0.185·29-s − 1.79·31-s − 0.176·32-s − 0.171·34-s + 1.31·37-s + 0.324·38-s + 0.624·41-s + 1.21·43-s − 0.452·44-s + 1.32·46-s + 1.45·47-s − 49-s + 0.131·58-s − 1.30·59-s + 0.896·61-s + 1.27·62-s + 1/8·64-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6848046629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6848046629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.27846425997226, −14.12460252161576, −13.14870352338026, −12.83173983285075, −12.37493565017931, −11.74892194537850, −11.08907378343905, −10.86073886227003, −10.15026279125088, −9.817522289398769, −9.206063121200216, −8.693545796080980, −8.057900331296574, −7.608960077246189, −7.311961683670292, −6.437113114580951, −5.769851698964970, −5.645868046470459, −4.616867973443189, −4.052736068821383, −3.404301586129508, −2.489606891328351, −2.175477401555817, −1.300103944010633, −0.3221572366469908,
0.3221572366469908, 1.300103944010633, 2.175477401555817, 2.489606891328351, 3.404301586129508, 4.052736068821383, 4.616867973443189, 5.645868046470459, 5.769851698964970, 6.437113114580951, 7.311961683670292, 7.608960077246189, 8.057900331296574, 8.693545796080980, 9.206063121200216, 9.817522289398769, 10.15026279125088, 10.86073886227003, 11.08907378343905, 11.74892194537850, 12.37493565017931, 12.83173983285075, 13.14870352338026, 14.12460252161576, 14.27846425997226
