| L(s) = 1 | − 2.31·2-s − 3-s + 3.35·4-s + 3.52·5-s + 2.31·6-s + 3.16·7-s − 3.13·8-s + 9-s − 8.15·10-s − 5.90·11-s − 3.35·12-s − 13-s − 7.31·14-s − 3.52·15-s + 0.550·16-s − 3.61·17-s − 2.31·18-s + 2.30·19-s + 11.8·20-s − 3.16·21-s + 13.6·22-s + 1.75·23-s + 3.13·24-s + 7.40·25-s + 2.31·26-s − 27-s + 10.6·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 0.577·3-s + 1.67·4-s + 1.57·5-s + 0.944·6-s + 1.19·7-s − 1.10·8-s + 0.333·9-s − 2.57·10-s − 1.78·11-s − 0.968·12-s − 0.277·13-s − 1.95·14-s − 0.909·15-s + 0.137·16-s − 0.877·17-s − 0.545·18-s + 0.527·19-s + 2.64·20-s − 0.690·21-s + 2.91·22-s + 0.366·23-s + 0.640·24-s + 1.48·25-s + 0.453·26-s − 0.192·27-s + 2.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 5 | \( 1 - 3.52T + 5T^{2} \) |
| 7 | \( 1 - 3.16T + 7T^{2} \) |
| 11 | \( 1 + 5.90T + 11T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 + 4.46T + 29T^{2} \) |
| 31 | \( 1 + 1.71T + 31T^{2} \) |
| 37 | \( 1 - 1.87T + 37T^{2} \) |
| 41 | \( 1 + 2.36T + 41T^{2} \) |
| 43 | \( 1 + 4.22T + 43T^{2} \) |
| 47 | \( 1 + 7.03T + 47T^{2} \) |
| 53 | \( 1 - 3.43T + 53T^{2} \) |
| 59 | \( 1 + 0.706T + 59T^{2} \) |
| 61 | \( 1 - 5.79T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 5.64T + 71T^{2} \) |
| 73 | \( 1 + 1.60T + 73T^{2} \) |
| 79 | \( 1 + 2.91T + 79T^{2} \) |
| 83 | \( 1 + 2.21T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 4.18T + 97T^{2} \) |
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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
−8.176038313034432782223647633873, −7.49669337179102958029868230791, −6.86752475003844687751523676676, −5.90513438841006192165182574580, −5.25364835751228465313208665748, −4.69800450410068865760371201458, −2.74858970834520673699174184896, −2.02277795435236992944769000012, −1.39174271319411748173652707501, 0,
1.39174271319411748173652707501, 2.02277795435236992944769000012, 2.74858970834520673699174184896, 4.69800450410068865760371201458, 5.25364835751228465313208665748, 5.90513438841006192165182574580, 6.86752475003844687751523676676, 7.49669337179102958029868230791, 8.176038313034432782223647633873
