| L(s) = 1 | − 1.09·2-s + 2.75·3-s − 0.810·4-s + 1.35·5-s − 3.00·6-s + 0.0387·7-s + 3.06·8-s + 4.60·9-s − 1.47·10-s − 2.52·11-s − 2.23·12-s + 3.42·13-s − 0.0422·14-s + 3.72·15-s − 1.72·16-s + 0.349·17-s − 5.01·18-s + 0.797·19-s − 1.09·20-s + 0.106·21-s + 2.75·22-s + 7.33·23-s + 8.45·24-s − 3.16·25-s − 3.73·26-s + 4.41·27-s − 0.0313·28-s + ⋯ |
| L(s) = 1 | − 0.771·2-s + 1.59·3-s − 0.405·4-s + 0.604·5-s − 1.22·6-s + 0.0146·7-s + 1.08·8-s + 1.53·9-s − 0.466·10-s − 0.761·11-s − 0.644·12-s + 0.950·13-s − 0.0112·14-s + 0.962·15-s − 0.430·16-s + 0.0848·17-s − 1.18·18-s + 0.182·19-s − 0.245·20-s + 0.0232·21-s + 0.587·22-s + 1.53·23-s + 1.72·24-s − 0.633·25-s − 0.732·26-s + 0.849·27-s − 0.00592·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.486765805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.486765805\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 349 | \( 1 - T \) |
| good | 2 | \( 1 + 1.09T + 2T^{2} \) |
| 3 | \( 1 - 2.75T + 3T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 7 | \( 1 - 0.0387T + 7T^{2} \) |
| 11 | \( 1 + 2.52T + 11T^{2} \) |
| 13 | \( 1 - 3.42T + 13T^{2} \) |
| 17 | \( 1 - 0.349T + 17T^{2} \) |
| 19 | \( 1 - 0.797T + 19T^{2} \) |
| 23 | \( 1 - 7.33T + 23T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 31 | \( 1 + 3.02T + 31T^{2} \) |
| 37 | \( 1 - 2.34T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 - 4.67T + 43T^{2} \) |
| 47 | \( 1 - 6.08T + 47T^{2} \) |
| 53 | \( 1 - 1.83T + 53T^{2} \) |
| 59 | \( 1 - 5.73T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + 5.42T + 67T^{2} \) |
| 71 | \( 1 + 9.26T + 71T^{2} \) |
| 73 | \( 1 + 1.57T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 6.51T + 83T^{2} \) |
| 89 | \( 1 + 0.0481T + 89T^{2} \) |
| 97 | \( 1 + 0.0462T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.07522380665999777690726644227, −10.20857482243863964884751811598, −9.302134132679386353600285328626, −8.897427578221351271480145078271, −7.976942172555191812511029105785, −7.27873301781998570240883235060, −5.59548521209339055595678409566, −4.18770139309976181744083858901, −2.95568821138191581129683847145, −1.58725570795479694378380430466,
1.58725570795479694378380430466, 2.95568821138191581129683847145, 4.18770139309976181744083858901, 5.59548521209339055595678409566, 7.27873301781998570240883235060, 7.976942172555191812511029105785, 8.897427578221351271480145078271, 9.302134132679386353600285328626, 10.20857482243863964884751811598, 11.07522380665999777690726644227
