| L(s) = 1 | + 2.43·2-s + 3.90·4-s − 3.43·5-s − 7-s + 4.64·8-s − 8.34·10-s + 6.11·13-s − 2.43·14-s + 3.46·16-s + 3.21·17-s − 6.34·19-s − 13.4·20-s + 0.210·23-s + 6.77·25-s + 14.8·26-s − 3.90·28-s + 2.74·29-s + 4.64·31-s − 0.861·32-s + 7.80·34-s + 3.43·35-s − 9.37·37-s − 15.4·38-s − 15.9·40-s + 11.9·41-s − 7.12·43-s + 0.511·46-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 1.95·4-s − 1.53·5-s − 0.377·7-s + 1.64·8-s − 2.63·10-s + 1.69·13-s − 0.649·14-s + 0.866·16-s + 0.778·17-s − 1.45·19-s − 2.99·20-s + 0.0438·23-s + 1.35·25-s + 2.91·26-s − 0.738·28-s + 0.509·29-s + 0.833·31-s − 0.152·32-s + 1.33·34-s + 0.579·35-s − 1.54·37-s − 2.50·38-s − 2.51·40-s + 1.86·41-s − 1.08·43-s + 0.0754·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.421428100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.421428100\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 5 | \( 1 + 3.43T + 5T^{2} \) |
| 13 | \( 1 - 6.11T + 13T^{2} \) |
| 17 | \( 1 - 3.21T + 17T^{2} \) |
| 19 | \( 1 + 6.34T + 19T^{2} \) |
| 23 | \( 1 - 0.210T + 23T^{2} \) |
| 29 | \( 1 - 2.74T + 29T^{2} \) |
| 31 | \( 1 - 4.64T + 31T^{2} \) |
| 37 | \( 1 + 9.37T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 7.12T + 43T^{2} \) |
| 47 | \( 1 - 5.91T + 47T^{2} \) |
| 53 | \( 1 - 6.35T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 9.20T + 61T^{2} \) |
| 67 | \( 1 + 3.89T + 67T^{2} \) |
| 71 | \( 1 - 5.19T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 4.33T + 79T^{2} \) |
| 83 | \( 1 + 0.737T + 83T^{2} \) |
| 89 | \( 1 - 9.34T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.68400837196936738133225455807, −6.92061491487405960903388418895, −6.34061440561548512445354961650, −5.77566940276320360073057326578, −4.83122637000696085352616898885, −4.18180333290181656561342726348, −3.64968970014405889627604218605, −3.24599389704391618927408378523, −2.18440187574299482339432440833, −0.805649533454790470065871473691,
0.805649533454790470065871473691, 2.18440187574299482339432440833, 3.24599389704391618927408378523, 3.64968970014405889627604218605, 4.18180333290181656561342726348, 4.83122637000696085352616898885, 5.77566940276320360073057326578, 6.34061440561548512445354961650, 6.92061491487405960903388418895, 7.68400837196936738133225455807
