| L(s) = 1 | − 3·2-s + 4-s + 5·5-s + 21·8-s − 15·10-s + 45·11-s − 59·13-s − 71·16-s − 54·17-s + 121·19-s + 5·20-s − 135·22-s − 69·23-s + 25·25-s + 177·26-s + 162·29-s + 88·31-s + 45·32-s + 162·34-s − 259·37-s − 363·38-s + 105·40-s + 195·41-s − 286·43-s + 45·44-s + 207·46-s + 45·47-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s + 0.447·5-s + 0.928·8-s − 0.474·10-s + 1.23·11-s − 1.25·13-s − 1.10·16-s − 0.770·17-s + 1.46·19-s + 0.0559·20-s − 1.30·22-s − 0.625·23-s + 1/5·25-s + 1.33·26-s + 1.03·29-s + 0.509·31-s + 0.248·32-s + 0.817·34-s − 1.15·37-s − 1.54·38-s + 0.415·40-s + 0.742·41-s − 1.01·43-s + 0.154·44-s + 0.663·46-s + 0.139·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 45 T + p^{3} T^{2} \) |
| 13 | \( 1 + 59 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 121 T + p^{3} T^{2} \) |
| 23 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 162 T + p^{3} T^{2} \) |
| 31 | \( 1 - 88 T + p^{3} T^{2} \) |
| 37 | \( 1 + 7 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 195 T + p^{3} T^{2} \) |
| 43 | \( 1 + 286 T + p^{3} T^{2} \) |
| 47 | \( 1 - 45 T + p^{3} T^{2} \) |
| 53 | \( 1 + 597 T + p^{3} T^{2} \) |
| 59 | \( 1 + 360 T + p^{3} T^{2} \) |
| 61 | \( 1 + 392 T + p^{3} T^{2} \) |
| 67 | \( 1 + 280 T + p^{3} T^{2} \) |
| 71 | \( 1 + 48 T + p^{3} T^{2} \) |
| 73 | \( 1 + 668 T + p^{3} T^{2} \) |
| 79 | \( 1 - 782 T + p^{3} T^{2} \) |
| 83 | \( 1 - 768 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1194 T + p^{3} T^{2} \) |
| 97 | \( 1 + 902 T + p^{3} T^{2} \) |
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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
−8.486360374528819730769977492192, −7.61980430977609205921360852507, −6.96902737809414454401184517056, −6.18791587901461618806730812667, −4.99529480653853273812618354095, −4.41722806738689428454547295696, −3.16005052111057680353406111268, −1.96912733217402228805118742041, −1.13737469287161189185703235592, 0,
1.13737469287161189185703235592, 1.96912733217402228805118742041, 3.16005052111057680353406111268, 4.41722806738689428454547295696, 4.99529480653853273812618354095, 6.18791587901461618806730812667, 6.96902737809414454401184517056, 7.61980430977609205921360852507, 8.486360374528819730769977492192
