| L(s) = 1 | + 2.35·2-s − 0.229·3-s + 3.52·4-s − 0.539·6-s + 2.06·7-s + 3.59·8-s − 2.94·9-s − 0.635·11-s − 0.808·12-s + 0.375·13-s + 4.85·14-s + 1.39·16-s − 6.93·18-s + 7.47·19-s − 0.473·21-s − 1.49·22-s + 8.34·23-s − 0.823·24-s + 0.883·26-s + 1.36·27-s + 7.29·28-s + 3.44·29-s + 6.73·31-s − 3.91·32-s + 0.145·33-s − 10.3·36-s + 4.01·37-s + ⋯ |
| L(s) = 1 | + 1.66·2-s − 0.132·3-s + 1.76·4-s − 0.220·6-s + 0.781·7-s + 1.27·8-s − 0.982·9-s − 0.191·11-s − 0.233·12-s + 0.104·13-s + 1.29·14-s + 0.347·16-s − 1.63·18-s + 1.71·19-s − 0.103·21-s − 0.318·22-s + 1.74·23-s − 0.168·24-s + 0.173·26-s + 0.262·27-s + 1.37·28-s + 0.638·29-s + 1.21·31-s − 0.691·32-s + 0.0253·33-s − 1.73·36-s + 0.659·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.914650700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.914650700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 3 | \( 1 + 0.229T + 3T^{2} \) |
| 7 | \( 1 - 2.06T + 7T^{2} \) |
| 11 | \( 1 + 0.635T + 11T^{2} \) |
| 13 | \( 1 - 0.375T + 13T^{2} \) |
| 19 | \( 1 - 7.47T + 19T^{2} \) |
| 23 | \( 1 - 8.34T + 23T^{2} \) |
| 29 | \( 1 - 3.44T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 - 4.01T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 9.36T + 43T^{2} \) |
| 47 | \( 1 + 7.98T + 47T^{2} \) |
| 53 | \( 1 - 2.38T + 53T^{2} \) |
| 59 | \( 1 - 0.380T + 59T^{2} \) |
| 61 | \( 1 - 7.77T + 61T^{2} \) |
| 67 | \( 1 - 6.63T + 67T^{2} \) |
| 71 | \( 1 + 0.277T + 71T^{2} \) |
| 73 | \( 1 - 9.98T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 9.85T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−7.85994386268815612003922432792, −6.77970331241132914629666165551, −6.50546037080160374505893924057, −5.35333215353343017477588181678, −5.16534740608627812367836963664, −4.64644514220091974278773428885, −3.37223310757804134123025196626, −3.13952986660679044008866627567, −2.17367670721986284632213628972, −0.993101471901926181675802169792,
0.993101471901926181675802169792, 2.17367670721986284632213628972, 3.13952986660679044008866627567, 3.37223310757804134123025196626, 4.64644514220091974278773428885, 5.16534740608627812367836963664, 5.35333215353343017477588181678, 6.50546037080160374505893924057, 6.77970331241132914629666165551, 7.85994386268815612003922432792
