| L(s) = 1 | + 2.00·2-s − 3·3-s − 3.98·4-s + 19.2·5-s − 6.00·6-s + 6.40·7-s − 24.0·8-s + 9·9-s + 38.5·10-s + 11·11-s + 11.9·12-s − 23.0·13-s + 12.8·14-s − 57.6·15-s − 16.1·16-s − 5.33·17-s + 18.0·18-s + 19.3·19-s − 76.7·20-s − 19.2·21-s + 22.0·22-s − 8.57·23-s + 72.0·24-s + 244.·25-s − 46.1·26-s − 27·27-s − 25.5·28-s + ⋯ |
| L(s) = 1 | + 0.708·2-s − 0.577·3-s − 0.498·4-s + 1.71·5-s − 0.408·6-s + 0.345·7-s − 1.06·8-s + 0.333·9-s + 1.21·10-s + 0.301·11-s + 0.287·12-s − 0.491·13-s + 0.244·14-s − 0.992·15-s − 0.252·16-s − 0.0760·17-s + 0.236·18-s + 0.234·19-s − 0.857·20-s − 0.199·21-s + 0.213·22-s − 0.0777·23-s + 0.612·24-s + 1.95·25-s − 0.347·26-s − 0.192·27-s − 0.172·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.143968380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.143968380\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 2.00T + 8T^{2} \) |
| 5 | \( 1 - 19.2T + 125T^{2} \) |
| 7 | \( 1 - 6.40T + 343T^{2} \) |
| 13 | \( 1 + 23.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.33T + 4.91e3T^{2} \) |
| 19 | \( 1 - 19.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 8.57T + 1.21e4T^{2} \) |
| 29 | \( 1 - 94.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 49.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 86.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 223.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 366.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 39.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 106.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 68.0T + 2.05e5T^{2} \) |
| 67 | \( 1 - 618.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 128.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 578.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 829.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 251.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 599.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 446.T + 9.12e5T^{2} \) |
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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
−9.073433201175063092871254050445, −8.066369061055551328049247636413, −6.80336558805981793377150725644, −6.21693364936773056083891461852, −5.42402828598730728431057573617, −5.02909361598227846723588783609, −4.10660095755814485177614168544, −2.87864909975536000956512417969, −1.89712563952337953719891979060, −0.77198955376676270755291943752,
0.77198955376676270755291943752, 1.89712563952337953719891979060, 2.87864909975536000956512417969, 4.10660095755814485177614168544, 5.02909361598227846723588783609, 5.42402828598730728431057573617, 6.21693364936773056083891461852, 6.80336558805981793377150725644, 8.066369061055551328049247636413, 9.073433201175063092871254050445
