| L(s) = 1 | + 3-s + 5-s + 2.82·7-s + 9-s + 4·11-s − 13-s + 15-s + 4.82·17-s − 2.82·19-s + 2.82·21-s + 2.82·23-s + 25-s + 27-s − 4.82·29-s − 9.65·31-s + 4·33-s + 2.82·35-s + 7.65·37-s − 39-s + 0.343·41-s + 9.65·43-s + 45-s − 1.65·47-s + 1.00·49-s + 4.82·51-s + 13.3·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.06·7-s + 0.333·9-s + 1.20·11-s − 0.277·13-s + 0.258·15-s + 1.17·17-s − 0.648·19-s + 0.617·21-s + 0.589·23-s + 0.200·25-s + 0.192·27-s − 0.896·29-s − 1.73·31-s + 0.696·33-s + 0.478·35-s + 1.25·37-s − 0.160·39-s + 0.0535·41-s + 1.47·43-s + 0.149·45-s − 0.241·47-s + 0.142·49-s + 0.676·51-s + 1.82·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.683580558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.683580558\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 9.65T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 - 0.343T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 8.82T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + 7.31T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.045336537735438964538464793851, −7.40608660327515587923515963424, −6.79782776734303638492618536007, −5.77632106967810826960549814806, −5.28310946296286182956009911966, −4.24192428795844122133685614619, −3.76668446133460394256597881550, −2.64757613569474325936206174357, −1.81103089310279446695566810905, −1.07321031752397014091573667401,
1.07321031752397014091573667401, 1.81103089310279446695566810905, 2.64757613569474325936206174357, 3.76668446133460394256597881550, 4.24192428795844122133685614619, 5.28310946296286182956009911966, 5.77632106967810826960549814806, 6.79782776734303638492618536007, 7.40608660327515587923515963424, 8.045336537735438964538464793851
