| L(s) = 1 | − 3-s + 1.90·5-s − 3.61·7-s + 9-s + 5.57·11-s + 5.61·13-s − 1.90·15-s + 6.17·17-s − 0.419·19-s + 3.61·21-s + 3.21·23-s − 1.37·25-s − 27-s + 3.62·29-s + 5.39·31-s − 5.57·33-s − 6.88·35-s − 2.73·37-s − 5.61·39-s − 2.39·41-s + 2.76·43-s + 1.90·45-s + 0.129·47-s + 6.09·49-s − 6.17·51-s + 8.82·53-s + 10.6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.851·5-s − 1.36·7-s + 0.333·9-s + 1.68·11-s + 1.55·13-s − 0.491·15-s + 1.49·17-s − 0.0961·19-s + 0.789·21-s + 0.670·23-s − 0.275·25-s − 0.192·27-s + 0.673·29-s + 0.968·31-s − 0.970·33-s − 1.16·35-s − 0.449·37-s − 0.899·39-s − 0.373·41-s + 0.421·43-s + 0.283·45-s + 0.0188·47-s + 0.871·49-s − 0.864·51-s + 1.21·53-s + 1.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.245930325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.245930325\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 - 1.90T + 5T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 11 | \( 1 - 5.57T + 11T^{2} \) |
| 13 | \( 1 - 5.61T + 13T^{2} \) |
| 17 | \( 1 - 6.17T + 17T^{2} \) |
| 19 | \( 1 + 0.419T + 19T^{2} \) |
| 23 | \( 1 - 3.21T + 23T^{2} \) |
| 29 | \( 1 - 3.62T + 29T^{2} \) |
| 31 | \( 1 - 5.39T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 - 2.76T + 43T^{2} \) |
| 47 | \( 1 - 0.129T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 + 0.0305T + 59T^{2} \) |
| 61 | \( 1 + 3.48T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 3.48T + 71T^{2} \) |
| 73 | \( 1 + 7.80T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 5.10T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.141889064389430901151162060667, −6.99660044970916908144430936892, −6.53024676848586301074658304149, −5.98707257223577008149715106250, −5.58380879687559635198248783196, −4.33579917956138740119302307197, −3.60656304306274836265319660672, −2.97369902137980433633331000960, −1.52909800837393569886928693935, −0.911381265721873903418492593212,
0.911381265721873903418492593212, 1.52909800837393569886928693935, 2.97369902137980433633331000960, 3.60656304306274836265319660672, 4.33579917956138740119302307197, 5.58380879687559635198248783196, 5.98707257223577008149715106250, 6.53024676848586301074658304149, 6.99660044970916908144430936892, 8.141889064389430901151162060667
