| L(s) = 1 | + 3.78·2-s − 8.20·3-s − 17.6·4-s − 31.0·6-s − 88.9·7-s − 188.·8-s − 175.·9-s − 156.·11-s + 145.·12-s − 169·13-s − 336.·14-s − 145.·16-s − 447.·17-s − 664.·18-s − 269.·19-s + 730.·21-s − 592.·22-s − 1.37e3·23-s + 1.54e3·24-s − 639.·26-s + 3.43e3·27-s + 1.57e3·28-s − 3.69e3·29-s + 797.·31-s + 5.46e3·32-s + 1.28e3·33-s − 1.69e3·34-s + ⋯ |
| L(s) = 1 | + 0.668·2-s − 0.526·3-s − 0.552·4-s − 0.352·6-s − 0.686·7-s − 1.03·8-s − 0.722·9-s − 0.390·11-s + 0.290·12-s − 0.277·13-s − 0.459·14-s − 0.142·16-s − 0.375·17-s − 0.483·18-s − 0.171·19-s + 0.361·21-s − 0.261·22-s − 0.540·23-s + 0.546·24-s − 0.185·26-s + 0.907·27-s + 0.379·28-s − 0.816·29-s + 0.149·31-s + 0.943·32-s + 0.205·33-s − 0.251·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7580191801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7580191801\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 169T \) |
| good | 2 | \( 1 - 3.78T + 32T^{2} \) |
| 3 | \( 1 + 8.20T + 243T^{2} \) |
| 7 | \( 1 + 88.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 156.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 447.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 269.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.37e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 797.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.39e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.43e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.13e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.97e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.15e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.13e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.21e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.30e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.54e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.73e4T + 8.58e9T^{2} \) |
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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
−10.93154635370962693217914533373, −9.819359874113385446827075130203, −9.006768629930498181630075410956, −7.947641408221659107676034296682, −6.50518251108565296809626075942, −5.75187768145654263500815590950, −4.85812672178506052188292407225, −3.70912869418540935032313904278, −2.58187854116373633571287599977, −0.42496638566140978183476951573,
0.42496638566140978183476951573, 2.58187854116373633571287599977, 3.70912869418540935032313904278, 4.85812672178506052188292407225, 5.75187768145654263500815590950, 6.50518251108565296809626075942, 7.947641408221659107676034296682, 9.006768629930498181630075410956, 9.819359874113385446827075130203, 10.93154635370962693217914533373
