L(s) = 1 | + 2-s + 2.47·3-s + 4-s + 5-s + 2.47·6-s + 2.13·7-s + 8-s + 3.13·9-s + 10-s + 4.47·11-s + 2.47·12-s + 0.137·13-s + 2.13·14-s + 2.47·15-s + 16-s + 1.52·17-s + 3.13·18-s + 5.09·19-s + 20-s + 5.29·21-s + 4.47·22-s + 2.47·24-s + 25-s + 0.137·26-s + 0.340·27-s + 2.13·28-s − 7.22·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.43·3-s + 0.5·4-s + 0.447·5-s + 1.01·6-s + 0.807·7-s + 0.353·8-s + 1.04·9-s + 0.316·10-s + 1.34·11-s + 0.715·12-s + 0.0380·13-s + 0.571·14-s + 0.639·15-s + 0.250·16-s + 0.369·17-s + 0.739·18-s + 1.16·19-s + 0.223·20-s + 1.15·21-s + 0.954·22-s + 0.505·24-s + 0.200·25-s + 0.0269·26-s + 0.0654·27-s + 0.403·28-s − 1.34·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.054635374\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.054635374\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 2.47T + 3T^{2} \) |
| 7 | \( 1 - 2.13T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 - 0.137T + 13T^{2} \) |
| 17 | \( 1 - 1.52T + 17T^{2} \) |
| 19 | \( 1 - 5.09T + 19T^{2} \) |
| 29 | \( 1 + 7.22T + 29T^{2} \) |
| 31 | \( 1 + 2.81T + 31T^{2} \) |
| 37 | \( 1 + 11.9T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 7.22T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 - 1.72T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 9.15T + 61T^{2} \) |
| 67 | \( 1 - 2.95T + 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 4.68T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 4.27T + 89T^{2} \) |
| 97 | \( 1 + 19.4T + 97T^{2} \) |
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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
−8.241853294773742296735205091662, −7.39100434811262087735260265132, −6.97977131620200437226204871285, −5.89887874168825603372319564766, −5.23779795035598359886424351412, −4.33838601913248892248371156637, −3.54044504110973782144627918519, −3.08833178563317813726101258943, −1.84390653122024934670579652928, −1.52263238377104808173335442210,
1.52263238377104808173335442210, 1.84390653122024934670579652928, 3.08833178563317813726101258943, 3.54044504110973782144627918519, 4.33838601913248892248371156637, 5.23779795035598359886424351412, 5.89887874168825603372319564766, 6.97977131620200437226204871285, 7.39100434811262087735260265132, 8.241853294773742296735205091662