| L(s) = 1 | + 1.59·3-s + 5-s − 4.50·7-s − 0.463·9-s + 5.01·11-s − 0.0423·13-s + 1.59·15-s − 3.74·17-s + 5.42·19-s − 7.17·21-s + 4.97·23-s + 25-s − 5.51·27-s + 4.97·29-s + 4.62·31-s + 7.98·33-s − 4.50·35-s − 37-s − 0.0674·39-s + 5.01·41-s − 0.112·43-s − 0.463·45-s + 1.60·47-s + 13.2·49-s − 5.96·51-s − 10.8·53-s + 5.01·55-s + ⋯ |
| L(s) = 1 | + 0.919·3-s + 0.447·5-s − 1.70·7-s − 0.154·9-s + 1.51·11-s − 0.0117·13-s + 0.411·15-s − 0.908·17-s + 1.24·19-s − 1.56·21-s + 1.03·23-s + 0.200·25-s − 1.06·27-s + 0.923·29-s + 0.830·31-s + 1.39·33-s − 0.761·35-s − 0.164·37-s − 0.0108·39-s + 0.783·41-s − 0.0171·43-s − 0.0690·45-s + 0.234·47-s + 1.89·49-s − 0.835·51-s − 1.49·53-s + 0.676·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.427340303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.427340303\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 1.59T + 3T^{2} \) |
| 7 | \( 1 + 4.50T + 7T^{2} \) |
| 11 | \( 1 - 5.01T + 11T^{2} \) |
| 13 | \( 1 + 0.0423T + 13T^{2} \) |
| 17 | \( 1 + 3.74T + 17T^{2} \) |
| 19 | \( 1 - 5.42T + 19T^{2} \) |
| 23 | \( 1 - 4.97T + 23T^{2} \) |
| 29 | \( 1 - 4.97T + 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 41 | \( 1 - 5.01T + 41T^{2} \) |
| 43 | \( 1 + 0.112T + 43T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 9.19T + 59T^{2} \) |
| 61 | \( 1 + 8.80T + 61T^{2} \) |
| 67 | \( 1 + 0.203T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 7.45T + 73T^{2} \) |
| 79 | \( 1 + 6.13T + 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 + 4.13T + 97T^{2} \) |
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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
−9.106306797431896505056875716339, −8.145818871767513569312603451342, −7.06741574233738803360175053058, −6.53804098740504834081467345659, −5.95465162516394125603627833713, −4.76268660268179950266234507624, −3.65036214397222232239846010032, −3.15117425739817709490130486258, −2.33804573798431334728166562072, −0.923568043368975551631184026104,
0.923568043368975551631184026104, 2.33804573798431334728166562072, 3.15117425739817709490130486258, 3.65036214397222232239846010032, 4.76268660268179950266234507624, 5.95465162516394125603627833713, 6.53804098740504834081467345659, 7.06741574233738803360175053058, 8.145818871767513569312603451342, 9.106306797431896505056875716339
