L(s) = 1 | − 0.515·2-s + 3.36·3-s − 1.73·4-s − 1.73·6-s − 7-s + 1.92·8-s + 8.30·9-s + 1.20·11-s − 5.83·12-s + 2.81·13-s + 0.515·14-s + 2.47·16-s − 1.95·17-s − 4.28·18-s − 0.107·19-s − 3.36·21-s − 0.623·22-s + 23-s + 6.47·24-s − 1.45·26-s + 17.8·27-s + 1.73·28-s + 0.846·29-s − 8.74·31-s − 5.12·32-s + 4.06·33-s + 1.00·34-s + ⋯ |
L(s) = 1 | − 0.364·2-s + 1.94·3-s − 0.867·4-s − 0.707·6-s − 0.377·7-s + 0.680·8-s + 2.76·9-s + 0.364·11-s − 1.68·12-s + 0.780·13-s + 0.137·14-s + 0.618·16-s − 0.473·17-s − 1.00·18-s − 0.0247·19-s − 0.733·21-s − 0.132·22-s + 0.208·23-s + 1.32·24-s − 0.284·26-s + 3.43·27-s + 0.327·28-s + 0.157·29-s − 1.57·31-s − 0.906·32-s + 0.707·33-s + 0.172·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.915940300\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.915940300\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 0.515T + 2T^{2} \) |
| 3 | \( 1 - 3.36T + 3T^{2} \) |
| 11 | \( 1 - 1.20T + 11T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 + 1.95T + 17T^{2} \) |
| 19 | \( 1 + 0.107T + 19T^{2} \) |
| 29 | \( 1 - 0.846T + 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 - 2.07T + 37T^{2} \) |
| 41 | \( 1 + 0.748T + 41T^{2} \) |
| 43 | \( 1 - 9.08T + 43T^{2} \) |
| 47 | \( 1 - 8.58T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 4.49T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 + 0.821T + 67T^{2} \) |
| 71 | \( 1 - 2.02T + 71T^{2} \) |
| 73 | \( 1 + 7.95T + 73T^{2} \) |
| 79 | \( 1 - 7.88T + 79T^{2} \) |
| 83 | \( 1 + 0.403T + 83T^{2} \) |
| 89 | \( 1 - 6.13T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.734706228748942844716978266573, −7.83464965946116970043708688619, −7.39617121147745836784400373469, −6.49898409423235683973067532130, −5.33332238024382813341342625499, −4.10487991707008634799709634304, −3.95400525474373600227272260158, −2.99884006898263828269258512739, −2.02605779941841189664988870961, −1.01040981636889861289167822214,
1.01040981636889861289167822214, 2.02605779941841189664988870961, 2.99884006898263828269258512739, 3.95400525474373600227272260158, 4.10487991707008634799709634304, 5.33332238024382813341342625499, 6.49898409423235683973067532130, 7.39617121147745836784400373469, 7.83464965946116970043708688619, 8.734706228748942844716978266573