L(s) = 1 | + 2.42·2-s + 3-s + 3.87·4-s + 5-s + 2.42·6-s + 2.87·7-s + 4.53·8-s + 9-s + 2.42·10-s − 2.11·11-s + 3.87·12-s − 3.48·13-s + 6.95·14-s + 15-s + 3.24·16-s + 1.38·17-s + 2.42·18-s + 6.32·19-s + 3.87·20-s + 2.87·21-s − 5.11·22-s + 4.53·24-s + 25-s − 8.44·26-s + 27-s + 11.1·28-s + 0.760·29-s + ⋯ |
L(s) = 1 | + 1.71·2-s + 0.577·3-s + 1.93·4-s + 0.447·5-s + 0.989·6-s + 1.08·7-s + 1.60·8-s + 0.333·9-s + 0.766·10-s − 0.636·11-s + 1.11·12-s − 0.966·13-s + 1.85·14-s + 0.258·15-s + 0.810·16-s + 0.336·17-s + 0.571·18-s + 1.45·19-s + 0.865·20-s + 0.626·21-s − 1.09·22-s + 0.925·24-s + 0.200·25-s − 1.65·26-s + 0.192·27-s + 2.10·28-s + 0.141·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.356315029\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.356315029\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 11 | \( 1 + 2.11T + 11T^{2} \) |
| 13 | \( 1 + 3.48T + 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 29 | \( 1 - 0.760T + 29T^{2} \) |
| 31 | \( 1 - 0.871T + 31T^{2} \) |
| 37 | \( 1 - 8.34T + 37T^{2} \) |
| 41 | \( 1 + 1.38T + 41T^{2} \) |
| 43 | \( 1 - 9.59T + 43T^{2} \) |
| 47 | \( 1 + 7.85T + 47T^{2} \) |
| 53 | \( 1 - 4.98T + 53T^{2} \) |
| 59 | \( 1 - 8.57T + 59T^{2} \) |
| 61 | \( 1 + 1.48T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−7.54334017559504133186410496272, −7.22643045221218113447668730828, −6.17366027713318207187083184898, −5.49940685932826617780221789804, −4.94805775991058539939926741716, −4.48952848752353694308995938356, −3.54742686965944367764022883970, −2.73725351874180895965363681540, −2.27861602187062043682800633875, −1.25020550139045750356896403107,
1.25020550139045750356896403107, 2.27861602187062043682800633875, 2.73725351874180895965363681540, 3.54742686965944367764022883970, 4.48952848752353694308995938356, 4.94805775991058539939926741716, 5.49940685932826617780221789804, 6.17366027713318207187083184898, 7.22643045221218113447668730828, 7.54334017559504133186410496272