| L(s) = 1 | − 1.17·2-s − 0.618·4-s − 1.23·5-s − 4.23·7-s + 3.07·8-s + 1.45·10-s + 3.61·11-s + 3.07·13-s + 4.97·14-s − 2.38·16-s − 2.47·17-s + 0.763·20-s − 4.25·22-s + 2.61·23-s − 3.47·25-s − 3.61·26-s + 2.61·28-s + 2.62·29-s − 1.17·31-s − 3.35·32-s + 2.90·34-s + 5.23·35-s − 6.43·37-s − 3.80·40-s − 9.23·41-s − 6.32·43-s − 2.23·44-s + ⋯ |
| L(s) = 1 | − 0.831·2-s − 0.309·4-s − 0.552·5-s − 1.60·7-s + 1.08·8-s + 0.459·10-s + 1.09·11-s + 0.853·13-s + 1.33·14-s − 0.595·16-s − 0.599·17-s + 0.170·20-s − 0.906·22-s + 0.545·23-s − 0.694·25-s − 0.709·26-s + 0.494·28-s + 0.488·29-s − 0.211·31-s − 0.593·32-s + 0.498·34-s + 0.885·35-s − 1.05·37-s − 0.601·40-s − 1.44·41-s − 0.964·43-s − 0.337·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5458390831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5458390831\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 23 | \( 1 - 2.61T + 23T^{2} \) |
| 29 | \( 1 - 2.62T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 6.43T + 37T^{2} \) |
| 41 | \( 1 + 9.23T + 41T^{2} \) |
| 43 | \( 1 + 6.32T + 43T^{2} \) |
| 47 | \( 1 - 2.52T + 47T^{2} \) |
| 53 | \( 1 + 0.449T + 53T^{2} \) |
| 59 | \( 1 - 2.80T + 59T^{2} \) |
| 61 | \( 1 + 0.527T + 61T^{2} \) |
| 67 | \( 1 + 0.171T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + 7.60T + 79T^{2} \) |
| 83 | \( 1 + 1.23T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−8.816217060966053809595288673211, −8.155999813890574184784456675996, −7.05373796297607679314113077555, −6.70797764904848183242644231069, −5.81896200764659372081626962469, −4.64111835044437602501066024273, −3.77688876224997120909478074939, −3.29169349192003804218500661148, −1.71874797027255595004442478233, −0.51072249730915189108889981223,
0.51072249730915189108889981223, 1.71874797027255595004442478233, 3.29169349192003804218500661148, 3.77688876224997120909478074939, 4.64111835044437602501066024273, 5.81896200764659372081626962469, 6.70797764904848183242644231069, 7.05373796297607679314113077555, 8.155999813890574184784456675996, 8.816217060966053809595288673211
