| L(s) = 1 | − 3-s − 5-s + 3.41·7-s + 9-s + 1.41·11-s + 4.24·13-s + 15-s + 2.82·17-s + 19-s − 3.41·21-s + 4.82·23-s + 25-s − 27-s + 2.24·29-s + 8.82·31-s − 1.41·33-s − 3.41·35-s − 7.07·37-s − 4.24·39-s + 2.24·41-s + 1.75·43-s − 45-s − 4.82·47-s + 4.65·49-s − 2.82·51-s − 12.4·53-s − 1.41·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.29·7-s + 0.333·9-s + 0.426·11-s + 1.17·13-s + 0.258·15-s + 0.685·17-s + 0.229·19-s − 0.745·21-s + 1.00·23-s + 0.200·25-s − 0.192·27-s + 0.416·29-s + 1.58·31-s − 0.246·33-s − 0.577·35-s − 1.16·37-s − 0.679·39-s + 0.350·41-s + 0.267·43-s − 0.149·45-s − 0.704·47-s + 0.665·49-s − 0.396·51-s − 1.71·53-s − 0.190·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.093898191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.093898191\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 31 | \( 1 - 8.82T + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 + 4.82T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−8.232734835395951866786952815297, −7.73372965530860845693000267229, −6.80101487378957457616175655600, −6.19425871892741556483277878447, −5.23025417682103031654133567100, −4.73638373518931480150586112572, −3.89446926439897743578843011575, −3.01785667320687823454346050223, −1.58748057196747311901682976967, −0.937794634178985329137266693806,
0.937794634178985329137266693806, 1.58748057196747311901682976967, 3.01785667320687823454346050223, 3.89446926439897743578843011575, 4.73638373518931480150586112572, 5.23025417682103031654133567100, 6.19425871892741556483277878447, 6.80101487378957457616175655600, 7.73372965530860845693000267229, 8.232734835395951866786952815297
