| L(s) = 1 | + 2.06·2-s − 3.16·3-s + 2.27·4-s − 6.55·6-s − 3.93·7-s + 0.566·8-s + 7.04·9-s − 1.10·11-s − 7.20·12-s − 6.41·13-s − 8.14·14-s − 3.37·16-s − 7.11·17-s + 14.5·18-s + 0.694·19-s + 12.4·21-s − 2.28·22-s + 7.97·23-s − 1.79·24-s − 13.2·26-s − 12.8·27-s − 8.95·28-s − 9.53·29-s − 7.06·31-s − 8.11·32-s + 3.50·33-s − 14.7·34-s + ⋯ |
| L(s) = 1 | + 1.46·2-s − 1.82·3-s + 1.13·4-s − 2.67·6-s − 1.48·7-s + 0.200·8-s + 2.34·9-s − 0.333·11-s − 2.08·12-s − 1.77·13-s − 2.17·14-s − 0.844·16-s − 1.72·17-s + 3.43·18-s + 0.159·19-s + 2.72·21-s − 0.487·22-s + 1.66·23-s − 0.366·24-s − 2.60·26-s − 2.46·27-s − 1.69·28-s − 1.77·29-s − 1.26·31-s − 1.43·32-s + 0.610·33-s − 2.52·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3696799864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3696799864\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.06T + 2T^{2} \) |
| 3 | \( 1 + 3.16T + 3T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 + 6.41T + 13T^{2} \) |
| 17 | \( 1 + 7.11T + 17T^{2} \) |
| 19 | \( 1 - 0.694T + 19T^{2} \) |
| 23 | \( 1 - 7.97T + 23T^{2} \) |
| 29 | \( 1 + 9.53T + 29T^{2} \) |
| 31 | \( 1 + 7.06T + 31T^{2} \) |
| 37 | \( 1 - 5.67T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 + 3.80T + 43T^{2} \) |
| 47 | \( 1 + 1.99T + 47T^{2} \) |
| 53 | \( 1 - 6.08T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 - 6.40T + 61T^{2} \) |
| 67 | \( 1 + 0.552T + 67T^{2} \) |
| 71 | \( 1 - 0.706T + 71T^{2} \) |
| 73 | \( 1 - 8.09T + 73T^{2} \) |
| 79 | \( 1 + 9.29T + 79T^{2} \) |
| 83 | \( 1 + 5.06T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 5.95T + 97T^{2} \) |
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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
−7.37947032290128295211654793834, −6.80630599556698597855746710793, −6.67831785292817553376101047075, −5.55636820083121127626520937359, −5.40248669560590940660882765090, −4.60878963663915922102536133225, −3.99210686597280428106297421099, −3.00144410322708179365683100640, −2.11532755313416724273825362746, −0.25805051753955627516865262573,
0.25805051753955627516865262573, 2.11532755313416724273825362746, 3.00144410322708179365683100640, 3.99210686597280428106297421099, 4.60878963663915922102536133225, 5.40248669560590940660882765090, 5.55636820083121127626520937359, 6.67831785292817553376101047075, 6.80630599556698597855746710793, 7.37947032290128295211654793834
