| L(s) = 1 | + 18.4·2-s − 170.·4-s + 2.36e3·5-s + 7.02e3·7-s − 1.26e4·8-s + 4.36e4·10-s + 5.90e4·11-s − 8.40e4·13-s + 1.29e5·14-s − 1.46e5·16-s − 1.99e5·17-s + 2.93e5·19-s − 4.02e5·20-s + 1.09e6·22-s − 3.66e5·23-s + 3.63e6·25-s − 1.55e6·26-s − 1.19e6·28-s + 1.87e6·29-s + 3.91e6·31-s + 3.75e6·32-s − 3.68e6·34-s + 1.66e7·35-s + 2.22e7·37-s + 5.42e6·38-s − 2.98e7·40-s − 3.33e6·41-s + ⋯ |
| L(s) = 1 | + 0.817·2-s − 0.332·4-s + 1.69·5-s + 1.10·7-s − 1.08·8-s + 1.38·10-s + 1.21·11-s − 0.815·13-s + 0.903·14-s − 0.557·16-s − 0.578·17-s + 0.516·19-s − 0.562·20-s + 0.994·22-s − 0.273·23-s + 1.85·25-s − 0.666·26-s − 0.367·28-s + 0.492·29-s + 0.761·31-s + 0.633·32-s − 0.472·34-s + 1.87·35-s + 1.94·37-s + 0.422·38-s − 1.84·40-s − 0.184·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.515311194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.515311194\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 3.41e6T \) |
| good | 2 | \( 1 - 18.4T + 512T^{2} \) |
| 5 | \( 1 - 2.36e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 7.02e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.90e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 8.40e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.99e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.93e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.66e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.87e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.91e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.22e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.33e6T + 3.27e14T^{2} \) |
| 47 | \( 1 - 4.54e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.88e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.66e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.65e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.13e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.09e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 7.39e6T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.09e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.59e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.28e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.54e8T + 7.60e17T^{2} \) |
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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
−9.489957615788243738953281342852, −9.274953334552097282939410182265, −8.028446279124347492579577919883, −6.56989136548637484106181954604, −5.90855940009730303280516015306, −4.93604431820796447662440518876, −4.36156889903281977438881887104, −2.85825691941269623273874988653, −1.89855640459174876156387685796, −0.917802297931196132063024119858,
0.917802297931196132063024119858, 1.89855640459174876156387685796, 2.85825691941269623273874988653, 4.36156889903281977438881887104, 4.93604431820796447662440518876, 5.90855940009730303280516015306, 6.56989136548637484106181954604, 8.028446279124347492579577919883, 9.274953334552097282939410182265, 9.489957615788243738953281342852
