| L(s) = 1 | − 3·3-s − 5·5-s + 21.4·7-s + 9·9-s + 40.0·11-s − 13.4·13-s + 15·15-s + 118.·17-s + 81.5·19-s − 64.4·21-s + 23·23-s + 25·25-s − 27·27-s − 98.5·29-s + 41.9·31-s − 120.·33-s − 107.·35-s − 60.8·37-s + 40.4·39-s + 38.6·41-s + 141.·43-s − 45·45-s − 212.·47-s + 119.·49-s − 355.·51-s − 290.·53-s − 200.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.16·7-s + 0.333·9-s + 1.09·11-s − 0.287·13-s + 0.258·15-s + 1.69·17-s + 0.984·19-s − 0.670·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s − 0.631·29-s + 0.242·31-s − 0.633·33-s − 0.519·35-s − 0.270·37-s + 0.166·39-s + 0.147·41-s + 0.501·43-s − 0.149·45-s − 0.660·47-s + 0.347·49-s − 0.976·51-s − 0.754·53-s − 0.490·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.247682131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.247682131\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 7 | \( 1 - 21.4T + 343T^{2} \) |
| 11 | \( 1 - 40.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 13.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 81.5T + 6.85e3T^{2} \) |
| 29 | \( 1 + 98.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 41.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 60.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 38.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 212.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 290.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 734.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 322.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 833.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 780.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 201.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 6.04T + 4.93e5T^{2} \) |
| 83 | \( 1 - 195.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 224.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 877.T + 9.12e5T^{2} \) |
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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
−9.300073946571935403844481177732, −8.231124118705918252550319892121, −7.60925056606449901380435846465, −6.86835623530015215231214830823, −5.72597190946785200038977599980, −5.08507983663948270306416486277, −4.16966901964803224488321262121, −3.23478314426151298737099291495, −1.63434602044310504864892806750, −0.846096578006152281453282543995,
0.846096578006152281453282543995, 1.63434602044310504864892806750, 3.23478314426151298737099291495, 4.16966901964803224488321262121, 5.08507983663948270306416486277, 5.72597190946785200038977599980, 6.86835623530015215231214830823, 7.60925056606449901380435846465, 8.231124118705918252550319892121, 9.300073946571935403844481177732
