| L(s) = 1 | + 2.52·2-s + 0.772·3-s + 4.38·4-s + 5-s + 1.95·6-s − 7-s + 6.02·8-s − 2.40·9-s + 2.52·10-s − 4.65·11-s + 3.38·12-s − 2.02·13-s − 2.52·14-s + 0.772·15-s + 6.45·16-s − 3.04·17-s − 6.07·18-s − 5.44·19-s + 4.38·20-s − 0.772·21-s − 11.7·22-s − 3.23·23-s + 4.65·24-s + 25-s − 5.12·26-s − 4.17·27-s − 4.38·28-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 0.446·3-s + 2.19·4-s + 0.447·5-s + 0.797·6-s − 0.377·7-s + 2.12·8-s − 0.800·9-s + 0.799·10-s − 1.40·11-s + 0.977·12-s − 0.562·13-s − 0.675·14-s + 0.199·15-s + 1.61·16-s − 0.739·17-s − 1.43·18-s − 1.24·19-s + 0.980·20-s − 0.168·21-s − 2.50·22-s − 0.674·23-s + 0.950·24-s + 0.200·25-s − 1.00·26-s − 0.803·27-s − 0.828·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 3 | \( 1 - 0.772T + 3T^{2} \) |
| 11 | \( 1 + 4.65T + 11T^{2} \) |
| 13 | \( 1 + 2.02T + 13T^{2} \) |
| 17 | \( 1 + 3.04T + 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 + 7.17T + 29T^{2} \) |
| 31 | \( 1 + 0.258T + 31T^{2} \) |
| 37 | \( 1 - 8.98T + 37T^{2} \) |
| 41 | \( 1 - 3.70T + 41T^{2} \) |
| 43 | \( 1 - 6.91T + 43T^{2} \) |
| 47 | \( 1 - 6.76T + 47T^{2} \) |
| 53 | \( 1 + 7.19T + 53T^{2} \) |
| 59 | \( 1 - 2.52T + 59T^{2} \) |
| 61 | \( 1 + 6.93T + 61T^{2} \) |
| 67 | \( 1 + 6.80T + 67T^{2} \) |
| 71 | \( 1 - 4.04T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 4.58T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 5.64T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−7.46503720960629518489787029841, −6.37167387464189157178927635168, −5.97271538824006917901787092181, −5.41234313687278137127174394128, −4.60579189735325309924425345097, −4.01191161545390183366987418442, −3.07435442748679583656818827235, −2.41948564562235786206205311159, −2.13243104711440930655036518434, 0,
2.13243104711440930655036518434, 2.41948564562235786206205311159, 3.07435442748679583656818827235, 4.01191161545390183366987418442, 4.60579189735325309924425345097, 5.41234313687278137127174394128, 5.97271538824006917901787092181, 6.37167387464189157178927635168, 7.46503720960629518489787029841
