| L(s) = 1 | + 2.49·3-s + 0.946·5-s + 4.74·7-s + 3.24·9-s − 1.05·11-s − 0.242·13-s + 2.36·15-s − 2.74·17-s − 5.79·19-s + 11.8·21-s + 23-s − 4.10·25-s + 0.606·27-s + 1.86·29-s − 10.0·31-s − 2.63·33-s + 4.48·35-s + 0.946·37-s − 0.606·39-s + 6.35·41-s − 7.20·43-s + 3.06·45-s + 9.09·47-s + 15.4·49-s − 6.84·51-s + 12.7·53-s − 0.997·55-s + ⋯ |
| L(s) = 1 | + 1.44·3-s + 0.423·5-s + 1.79·7-s + 1.08·9-s − 0.317·11-s − 0.0673·13-s + 0.610·15-s − 0.664·17-s − 1.32·19-s + 2.58·21-s + 0.208·23-s − 0.821·25-s + 0.116·27-s + 0.346·29-s − 1.81·31-s − 0.458·33-s + 0.758·35-s + 0.155·37-s − 0.0971·39-s + 0.991·41-s − 1.09·43-s + 0.457·45-s + 1.32·47-s + 2.21·49-s − 0.959·51-s + 1.75·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.881311305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.881311305\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2.49T + 3T^{2} \) |
| 5 | \( 1 - 0.946T + 5T^{2} \) |
| 7 | \( 1 - 4.74T + 7T^{2} \) |
| 11 | \( 1 + 1.05T + 11T^{2} \) |
| 13 | \( 1 + 0.242T + 13T^{2} \) |
| 17 | \( 1 + 2.74T + 17T^{2} \) |
| 19 | \( 1 + 5.79T + 19T^{2} \) |
| 29 | \( 1 - 1.86T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 0.946T + 37T^{2} \) |
| 41 | \( 1 - 6.35T + 41T^{2} \) |
| 43 | \( 1 + 7.20T + 43T^{2} \) |
| 47 | \( 1 - 9.09T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 4.51T + 59T^{2} \) |
| 61 | \( 1 + 8.79T + 61T^{2} \) |
| 67 | \( 1 + 8.42T + 67T^{2} \) |
| 71 | \( 1 + 4.87T + 71T^{2} \) |
| 73 | \( 1 - 2.64T + 73T^{2} \) |
| 79 | \( 1 - 6.59T + 79T^{2} \) |
| 83 | \( 1 - 6.05T + 83T^{2} \) |
| 89 | \( 1 - 6.48T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−10.42261198646811766358346992202, −9.147278698187117909876618888836, −8.724059486248281566398624044794, −7.911181717165731819481081260199, −7.30485564769527841182113470082, −5.85851504606553132186991391047, −4.73930210828356443510637895691, −3.88704973625477032816960460303, −2.38763060486164376241986010252, −1.81600144651494457716281538431,
1.81600144651494457716281538431, 2.38763060486164376241986010252, 3.88704973625477032816960460303, 4.73930210828356443510637895691, 5.85851504606553132186991391047, 7.30485564769527841182113470082, 7.911181717165731819481081260199, 8.724059486248281566398624044794, 9.147278698187117909876618888836, 10.42261198646811766358346992202
