| L(s) = 1 | − 2.90·3-s + 5-s − 2.18·7-s + 5.43·9-s + 6.16·13-s − 2.90·15-s + 7.12·17-s + 6.35·19-s + 6.34·21-s + 2.17·23-s + 25-s − 7.08·27-s − 3.19·29-s + 4.39·31-s − 2.18·35-s + 5.29·37-s − 17.8·39-s + 3.28·41-s − 6.83·43-s + 5.43·45-s − 4.30·47-s − 2.23·49-s − 20.7·51-s − 10.6·53-s − 18.4·57-s + 13.7·59-s + 7.35·61-s + ⋯ |
| L(s) = 1 | − 1.67·3-s + 0.447·5-s − 0.825·7-s + 1.81·9-s + 1.70·13-s − 0.750·15-s + 1.72·17-s + 1.45·19-s + 1.38·21-s + 0.454·23-s + 0.200·25-s − 1.36·27-s − 0.592·29-s + 0.789·31-s − 0.369·35-s + 0.870·37-s − 2.86·39-s + 0.513·41-s − 1.04·43-s + 0.810·45-s − 0.627·47-s − 0.319·49-s − 2.89·51-s − 1.45·53-s − 2.44·57-s + 1.79·59-s + 0.941·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.343838400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.343838400\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 7 | \( 1 + 2.18T + 7T^{2} \) |
| 13 | \( 1 - 6.16T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 - 6.35T + 19T^{2} \) |
| 23 | \( 1 - 2.17T + 23T^{2} \) |
| 29 | \( 1 + 3.19T + 29T^{2} \) |
| 31 | \( 1 - 4.39T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 - 3.28T + 41T^{2} \) |
| 43 | \( 1 + 6.83T + 43T^{2} \) |
| 47 | \( 1 + 4.30T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 7.35T + 61T^{2} \) |
| 67 | \( 1 + 7.67T + 67T^{2} \) |
| 71 | \( 1 - 5.74T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 + 8.11T + 79T^{2} \) |
| 83 | \( 1 - 4.22T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 3.27T + 97T^{2} \) |
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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
−8.150630791807651184329842195423, −7.34759984367217376714953878639, −6.50686475832865110641552235686, −6.06984137001283276462087295102, −5.51842045142095960501867792065, −4.90041363300666020018844243059, −3.71561115108025632176880894706, −3.10863208359682726634040636444, −1.39095870833861698773649783696, −0.803143487977651595601699749608,
0.803143487977651595601699749608, 1.39095870833861698773649783696, 3.10863208359682726634040636444, 3.71561115108025632176880894706, 4.90041363300666020018844243059, 5.51842045142095960501867792065, 6.06984137001283276462087295102, 6.50686475832865110641552235686, 7.34759984367217376714953878639, 8.150630791807651184329842195423
