| L(s) = 1 | − 1.13·2-s − 0.718·4-s − 4.11·7-s + 3.07·8-s + 5.78·11-s + 6.78·13-s + 4.65·14-s − 2.04·16-s + 5.41·17-s − 19-s − 6.54·22-s + 8.04·23-s − 7.67·26-s + 2.95·28-s − 5.34·29-s + 0.327·31-s − 3.83·32-s − 6.12·34-s + 10.7·37-s + 1.13·38-s + 2.70·41-s + 0.654·43-s − 4.15·44-s − 9.11·46-s − 7.67·47-s + 9.89·49-s − 4.87·52-s + ⋯ |
| L(s) = 1 | − 0.800·2-s − 0.359·4-s − 1.55·7-s + 1.08·8-s + 1.74·11-s + 1.88·13-s + 1.24·14-s − 0.511·16-s + 1.31·17-s − 0.229·19-s − 1.39·22-s + 1.67·23-s − 1.50·26-s + 0.558·28-s − 0.991·29-s + 0.0587·31-s − 0.678·32-s − 1.05·34-s + 1.77·37-s + 0.183·38-s + 0.422·41-s + 0.0998·43-s − 0.626·44-s − 1.34·46-s − 1.11·47-s + 1.41·49-s − 0.676·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.219474526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.219474526\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.13T + 2T^{2} \) |
| 7 | \( 1 + 4.11T + 7T^{2} \) |
| 11 | \( 1 - 5.78T + 11T^{2} \) |
| 13 | \( 1 - 6.78T + 13T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 23 | \( 1 - 8.04T + 23T^{2} \) |
| 29 | \( 1 + 5.34T + 29T^{2} \) |
| 31 | \( 1 - 0.327T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 - 0.654T + 43T^{2} \) |
| 47 | \( 1 + 7.67T + 47T^{2} \) |
| 53 | \( 1 + 0.813T + 53T^{2} \) |
| 59 | \( 1 + 4.97T + 59T^{2} \) |
| 61 | \( 1 + 7.87T + 61T^{2} \) |
| 67 | \( 1 - 9.76T + 67T^{2} \) |
| 71 | \( 1 - 4.97T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 1.01T + 79T^{2} \) |
| 83 | \( 1 + 1.25T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 6.78T + 97T^{2} \) |
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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
−8.645559630602223469002747658667, −7.77319371030302314620335320314, −6.92840296431310345324356536107, −6.28981967398762436247361419850, −5.72587740312920340713143082154, −4.42371175340781856095559766199, −3.65898852283740976653365637813, −3.18827542961897747808853792720, −1.42653737968929985886927062712, −0.817843743570779719769716503409,
0.817843743570779719769716503409, 1.42653737968929985886927062712, 3.18827542961897747808853792720, 3.65898852283740976653365637813, 4.42371175340781856095559766199, 5.72587740312920340713143082154, 6.28981967398762436247361419850, 6.92840296431310345324356536107, 7.77319371030302314620335320314, 8.645559630602223469002747658667
