| L(s) = 1 | − 2-s + 4-s − 8-s + 5·11-s + 5·13-s + 16-s + 5·17-s + 5·19-s − 5·22-s + 3·23-s − 5·26-s + 29-s − 7·31-s − 32-s − 5·34-s + 11·37-s − 5·38-s − 6·41-s + 8·43-s + 5·44-s − 3·46-s + 2·47-s − 7·49-s + 5·52-s − 10·53-s − 58-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.50·11-s + 1.38·13-s + 1/4·16-s + 1.21·17-s + 1.14·19-s − 1.06·22-s + 0.625·23-s − 0.980·26-s + 0.185·29-s − 1.25·31-s − 0.176·32-s − 0.857·34-s + 1.80·37-s − 0.811·38-s − 0.937·41-s + 1.21·43-s + 0.753·44-s − 0.442·46-s + 0.291·47-s − 49-s + 0.693·52-s − 1.37·53-s − 0.131·58-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.798437239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.798437239\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.15108686575358, −13.64781553411536, −13.00201991469186, −12.48134365752327, −11.95254074999013, −11.42412660571086, −11.11773075530750, −10.62927717797028, −9.816118791725664, −9.445063244115592, −9.150365438495437, −8.519004349417909, −7.908780919887990, −7.567356721358049, −6.754476943831510, −6.497698515042845, −5.716598745182807, −5.453359799156468, −4.438487842582139, −3.833203395631890, −3.318584992869943, −2.798039019832369, −1.610293498934585, −1.306489630724821, −0.7008471082744707,
0.7008471082744707, 1.306489630724821, 1.610293498934585, 2.798039019832369, 3.318584992869943, 3.833203395631890, 4.438487842582139, 5.453359799156468, 5.716598745182807, 6.497698515042845, 6.754476943831510, 7.567356721358049, 7.908780919887990, 8.519004349417909, 9.150365438495437, 9.445063244115592, 9.816118791725664, 10.62927717797028, 11.11773075530750, 11.42412660571086, 11.95254074999013, 12.48134365752327, 13.00201991469186, 13.64781553411536, 14.15108686575358
