| L(s) = 1 | − 2-s + 4-s − 5-s + 4·7-s − 8-s − 3·9-s + 10-s + 5·13-s − 4·14-s + 16-s − 3·17-s + 3·18-s − 19-s − 20-s + 4·23-s − 4·25-s − 5·26-s + 4·28-s − 3·29-s − 32-s + 3·34-s − 4·35-s − 3·36-s + 11·37-s + 38-s + 40-s − 3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s − 0.353·8-s − 9-s + 0.316·10-s + 1.38·13-s − 1.06·14-s + 1/4·16-s − 0.727·17-s + 0.707·18-s − 0.229·19-s − 0.223·20-s + 0.834·23-s − 4/5·25-s − 0.980·26-s + 0.755·28-s − 0.557·29-s − 0.176·32-s + 0.514·34-s − 0.676·35-s − 1/2·36-s + 1.80·37-s + 0.162·38-s + 0.158·40-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.416324971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.416324971\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.228912766309852102085870896595, −7.944497001586233269469134155298, −7.06659487479755281115064098579, −6.11307771952976772220682224994, −5.54434995099497542470554856367, −4.54404756217205910649773919421, −3.79466669921914109504870868868, −2.69213458982136888058863600164, −1.78324356096943383381539137364, −0.75586667936883934573547011091,
0.75586667936883934573547011091, 1.78324356096943383381539137364, 2.69213458982136888058863600164, 3.79466669921914109504870868868, 4.54404756217205910649773919421, 5.54434995099497542470554856367, 6.11307771952976772220682224994, 7.06659487479755281115064098579, 7.944497001586233269469134155298, 8.228912766309852102085870896595
