| L(s) = 1 | − 3.74·2-s − 2.12·3-s + 6.04·4-s − 10.5·5-s + 7.95·6-s + 7.32·8-s − 22.4·9-s + 39.4·10-s − 24.8·11-s − 12.8·12-s − 69.8·13-s + 22.3·15-s − 75.8·16-s + 75.4·17-s + 84.3·18-s − 127.·19-s − 63.6·20-s + 93.0·22-s − 24.6·23-s − 15.5·24-s − 14.1·25-s + 261.·26-s + 105.·27-s + 160.·29-s − 83.7·30-s + 43.6·31-s + 225.·32-s + ⋯ |
| L(s) = 1 | − 1.32·2-s − 0.408·3-s + 0.755·4-s − 0.941·5-s + 0.541·6-s + 0.323·8-s − 0.833·9-s + 1.24·10-s − 0.680·11-s − 0.308·12-s − 1.49·13-s + 0.384·15-s − 1.18·16-s + 1.07·17-s + 1.10·18-s − 1.53·19-s − 0.711·20-s + 0.901·22-s − 0.223·23-s − 0.132·24-s − 0.113·25-s + 1.97·26-s + 0.748·27-s + 1.02·29-s − 0.509·30-s + 0.253·31-s + 1.24·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| good | 2 | \( 1 + 3.74T + 8T^{2} \) |
| 3 | \( 1 + 2.12T + 27T^{2} \) |
| 5 | \( 1 + 10.5T + 125T^{2} \) |
| 11 | \( 1 + 24.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 69.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 75.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 24.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 43.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 169.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 393.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 241.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 418.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 149.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 182.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 968.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 364.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 493.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 858.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 552.T + 9.12e5T^{2} \) |
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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
−8.158068718904973726830471965590, −7.79389694062868238332859782659, −7.00608722139331270972275654314, −6.06952733566815938139524739148, −4.99832595269579530537857096611, −4.35320079282038246499212133930, −3.01866153484400863155099496607, −2.12220444633544283069486204795, −0.64630763748718656169125850022, 0,
0.64630763748718656169125850022, 2.12220444633544283069486204795, 3.01866153484400863155099496607, 4.35320079282038246499212133930, 4.99832595269579530537857096611, 6.06952733566815938139524739148, 7.00608722139331270972275654314, 7.79389694062868238332859782659, 8.158068718904973726830471965590
