| L(s) = 1 | − 2-s − 3.24·3-s + 4-s − 3.60·5-s + 3.24·6-s + 7-s − 8-s + 7.54·9-s + 3.60·10-s + 3.40·11-s − 3.24·12-s − 14-s + 11.7·15-s + 16-s − 3.80·17-s − 7.54·18-s + 6.26·19-s − 3.60·20-s − 3.24·21-s − 3.40·22-s − 1.50·23-s + 3.24·24-s + 7.98·25-s − 14.7·27-s + 28-s − 2.49·29-s − 11.7·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.87·3-s + 0.5·4-s − 1.61·5-s + 1.32·6-s + 0.377·7-s − 0.353·8-s + 2.51·9-s + 1.13·10-s + 1.02·11-s − 0.937·12-s − 0.267·14-s + 3.02·15-s + 0.250·16-s − 0.922·17-s − 1.77·18-s + 1.43·19-s − 0.805·20-s − 0.708·21-s − 0.726·22-s − 0.314·23-s + 0.662·24-s + 1.59·25-s − 2.83·27-s + 0.188·28-s − 0.463·29-s − 2.13·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3921760702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3921760702\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 3.24T + 3T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 11 | \( 1 - 3.40T + 11T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 - 6.26T + 19T^{2} \) |
| 23 | \( 1 + 1.50T + 23T^{2} \) |
| 29 | \( 1 + 2.49T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + 0.493T + 37T^{2} \) |
| 41 | \( 1 - 6.71T + 41T^{2} \) |
| 43 | \( 1 - 0.347T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 7.60T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 7.87T + 61T^{2} \) |
| 67 | \( 1 + 5.07T + 67T^{2} \) |
| 71 | \( 1 - 8.31T + 71T^{2} \) |
| 73 | \( 1 - 3.42T + 73T^{2} \) |
| 79 | \( 1 + 3.06T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 0.655T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−9.078337867245241376623372384965, −7.914568448415514337049921090770, −7.48160656790992205306720460696, −6.66374413186232704310891696210, −6.10291964046230962642282979925, −4.92149114361521989162646915390, −4.38804252798376160650083964378, −3.42950504575877212774771127464, −1.49255767680470477826450064297, −0.53085694387015859920448286151,
0.53085694387015859920448286151, 1.49255767680470477826450064297, 3.42950504575877212774771127464, 4.38804252798376160650083964378, 4.92149114361521989162646915390, 6.10291964046230962642282979925, 6.66374413186232704310891696210, 7.48160656790992205306720460696, 7.914568448415514337049921090770, 9.078337867245241376623372384965
