| L(s) = 1 | + 1.44·2-s + 0.0738·4-s − 0.734·5-s + 3.73·7-s − 2.77·8-s − 1.05·10-s + 2.70·11-s − 2.87·13-s + 5.38·14-s − 4.14·16-s + 2.75·17-s − 5.54·19-s − 0.0542·20-s + 3.89·22-s − 23-s − 4.46·25-s − 4.13·26-s + 0.275·28-s + 29-s − 10.2·31-s − 0.417·32-s + 3.96·34-s − 2.74·35-s − 11.4·37-s − 7.98·38-s + 2.03·40-s + 1.81·41-s + ⋯ |
| L(s) = 1 | + 1.01·2-s + 0.0369·4-s − 0.328·5-s + 1.41·7-s − 0.980·8-s − 0.334·10-s + 0.814·11-s − 0.796·13-s + 1.43·14-s − 1.03·16-s + 0.667·17-s − 1.27·19-s − 0.0121·20-s + 0.829·22-s − 0.208·23-s − 0.892·25-s − 0.811·26-s + 0.0521·28-s + 0.185·29-s − 1.84·31-s − 0.0737·32-s + 0.679·34-s − 0.463·35-s − 1.88·37-s − 1.29·38-s + 0.322·40-s + 0.282·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.44T + 2T^{2} \) |
| 5 | \( 1 + 0.734T + 5T^{2} \) |
| 7 | \( 1 - 3.73T + 7T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 - 2.75T + 17T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 1.81T + 41T^{2} \) |
| 43 | \( 1 - 0.181T + 43T^{2} \) |
| 47 | \( 1 - 6.02T + 47T^{2} \) |
| 53 | \( 1 - 0.662T + 53T^{2} \) |
| 59 | \( 1 - 0.174T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 - 15.9T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 0.990T + 73T^{2} \) |
| 79 | \( 1 + 2.96T + 79T^{2} \) |
| 83 | \( 1 + 1.08T + 83T^{2} \) |
| 89 | \( 1 + 5.21T + 89T^{2} \) |
| 97 | \( 1 + 3.97T + 97T^{2} \) |
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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
−7.62572409034832744764497500915, −7.01724359659210831741218613876, −6.05322975585703636448817824582, −5.37741874960961422311200934463, −4.81373575970864928654584798186, −4.04963968548128190858509127972, −3.64164481593169357488368369923, −2.38668308345754558024313329091, −1.57685552096810545096964700848, 0,
1.57685552096810545096964700848, 2.38668308345754558024313329091, 3.64164481593169357488368369923, 4.04963968548128190858509127972, 4.81373575970864928654584798186, 5.37741874960961422311200934463, 6.05322975585703636448817824582, 7.01724359659210831741218613876, 7.62572409034832744764497500915
