| L(s) = 1 | + 3-s − 0.475·5-s + 4.55·7-s + 9-s + 1.65·11-s − 0.475·15-s + 7.93·17-s + 2.60·19-s + 4.55·21-s + 3.81·23-s − 4.77·25-s + 27-s + 0.823·29-s + 1.79·31-s + 1.65·33-s − 2.16·35-s + 1.87·37-s − 10.7·41-s − 9.89·43-s − 0.475·45-s + 7.29·47-s + 13.7·49-s + 7.93·51-s − 10.0·53-s − 0.785·55-s + 2.60·57-s + 1.48·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.212·5-s + 1.72·7-s + 0.333·9-s + 0.498·11-s − 0.122·15-s + 1.92·17-s + 0.597·19-s + 0.994·21-s + 0.794·23-s − 0.954·25-s + 0.192·27-s + 0.152·29-s + 0.321·31-s + 0.287·33-s − 0.366·35-s + 0.307·37-s − 1.67·41-s − 1.50·43-s − 0.0708·45-s + 1.06·47-s + 1.96·49-s + 1.11·51-s − 1.37·53-s − 0.105·55-s + 0.344·57-s + 0.193·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.303628196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.303628196\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.475T + 5T^{2} \) |
| 7 | \( 1 - 4.55T + 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 17 | \( 1 - 7.93T + 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 - 3.81T + 23T^{2} \) |
| 29 | \( 1 - 0.823T + 29T^{2} \) |
| 31 | \( 1 - 1.79T + 31T^{2} \) |
| 37 | \( 1 - 1.87T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 9.89T + 43T^{2} \) |
| 47 | \( 1 - 7.29T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 1.48T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 3.32T + 67T^{2} \) |
| 71 | \( 1 + 1.90T + 71T^{2} \) |
| 73 | \( 1 + 6.14T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 7.30T + 89T^{2} \) |
| 97 | \( 1 - 4.18T + 97T^{2} \) |
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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
−8.329505799582283607733398127642, −7.72711399201349542855755154705, −7.36009158716880624088564437373, −6.20910001261660551886025054833, −5.23598069996423667753539532296, −4.76735745293299830478592044589, −3.75471387347871148142545742453, −3.05575391520615729915146249696, −1.77930874905384441784011183808, −1.16151269567537584482092650371,
1.16151269567537584482092650371, 1.77930874905384441784011183808, 3.05575391520615729915146249696, 3.75471387347871148142545742453, 4.76735745293299830478592044589, 5.23598069996423667753539532296, 6.20910001261660551886025054833, 7.36009158716880624088564437373, 7.72711399201349542855755154705, 8.329505799582283607733398127642
