| L(s) = 1 | + 4.78·2-s − 1.15·3-s + 14.8·4-s + 21.3·5-s − 5.53·6-s − 24.0·7-s + 32.7·8-s − 25.6·9-s + 101.·10-s + 17.5·11-s − 17.1·12-s − 56.9·13-s − 114.·14-s − 24.6·15-s + 37.8·16-s − 74.3·17-s − 122.·18-s − 100.·19-s + 316.·20-s + 27.8·21-s + 84.1·22-s − 13.9·23-s − 37.9·24-s + 328.·25-s − 272.·26-s + 60.9·27-s − 357.·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s − 0.222·3-s + 1.85·4-s + 1.90·5-s − 0.376·6-s − 1.29·7-s + 1.44·8-s − 0.950·9-s + 3.22·10-s + 0.482·11-s − 0.413·12-s − 1.21·13-s − 2.19·14-s − 0.424·15-s + 0.591·16-s − 1.06·17-s − 1.60·18-s − 1.21·19-s + 3.53·20-s + 0.289·21-s + 0.815·22-s − 0.126·23-s − 0.322·24-s + 2.63·25-s − 2.05·26-s + 0.434·27-s − 2.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 - 4.78T + 8T^{2} \) |
| 3 | \( 1 + 1.15T + 27T^{2} \) |
| 5 | \( 1 - 21.3T + 125T^{2} \) |
| 7 | \( 1 + 24.0T + 343T^{2} \) |
| 11 | \( 1 - 17.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 56.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 74.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 100.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 13.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 87.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 127.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 125.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 82.7T + 6.89e4T^{2} \) |
| 47 | \( 1 + 71.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 152.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 319.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 353.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 37.2T + 3.00e5T^{2} \) |
| 71 | \( 1 - 781.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.17e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 841.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 308.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.10e3T + 9.12e5T^{2} \) |
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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
−8.766719474470250813709058509457, −6.89119591867349598874489572926, −6.62978742149947101889743116097, −5.88837204771619008054739251580, −5.41642405707258241150737584502, −4.54045210985518340913860366622, −3.40068374200649222971859318730, −2.48835557367161954048494007245, −2.04756605907373287221339710053, 0,
2.04756605907373287221339710053, 2.48835557367161954048494007245, 3.40068374200649222971859318730, 4.54045210985518340913860366622, 5.41642405707258241150737584502, 5.88837204771619008054739251580, 6.62978742149947101889743116097, 6.89119591867349598874489572926, 8.766719474470250813709058509457
