| L(s) = 1 | − 0.795·3-s + 3.83·5-s + 0.297·7-s − 2.36·9-s − 2.28·11-s − 6.87·13-s − 3.05·15-s + 2.62·17-s − 2.89·19-s − 0.236·21-s + 3.98·23-s + 9.73·25-s + 4.26·27-s + 3.29·29-s − 3.05·31-s + 1.81·33-s + 1.14·35-s + 9.12·37-s + 5.47·39-s − 1.23·41-s − 5.27·43-s − 9.08·45-s − 13.5·47-s − 6.91·49-s − 2.08·51-s − 0.167·53-s − 8.76·55-s + ⋯ |
| L(s) = 1 | − 0.459·3-s + 1.71·5-s + 0.112·7-s − 0.789·9-s − 0.688·11-s − 1.90·13-s − 0.788·15-s + 0.636·17-s − 0.664·19-s − 0.0516·21-s + 0.831·23-s + 1.94·25-s + 0.821·27-s + 0.612·29-s − 0.549·31-s + 0.316·33-s + 0.193·35-s + 1.50·37-s + 0.875·39-s − 0.193·41-s − 0.804·43-s − 1.35·45-s − 1.98·47-s − 0.987·49-s − 0.292·51-s − 0.0229·53-s − 1.18·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 0.795T + 3T^{2} \) |
| 5 | \( 1 - 3.83T + 5T^{2} \) |
| 7 | \( 1 - 0.297T + 7T^{2} \) |
| 11 | \( 1 + 2.28T + 11T^{2} \) |
| 13 | \( 1 + 6.87T + 13T^{2} \) |
| 17 | \( 1 - 2.62T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 - 3.98T + 23T^{2} \) |
| 29 | \( 1 - 3.29T + 29T^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 - 9.12T + 37T^{2} \) |
| 41 | \( 1 + 1.23T + 41T^{2} \) |
| 43 | \( 1 + 5.27T + 43T^{2} \) |
| 47 | \( 1 + 13.5T + 47T^{2} \) |
| 53 | \( 1 + 0.167T + 53T^{2} \) |
| 59 | \( 1 - 0.470T + 59T^{2} \) |
| 61 | \( 1 + 4.07T + 61T^{2} \) |
| 67 | \( 1 + 9.63T + 67T^{2} \) |
| 71 | \( 1 - 2.27T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 1.77T + 83T^{2} \) |
| 89 | \( 1 - 8.24T + 89T^{2} \) |
| 97 | \( 1 + 6.79T + 97T^{2} \) |
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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
−8.093972413496771858570001723261, −7.24606783654665112200275371906, −6.40131822132186254897912965187, −5.85813553773490752096724667346, −4.99055362809910117649143767821, −4.86009205886332243916844869170, −2.97523965614867243658224775258, −2.54678619036393715711638393374, −1.52928380839761829735024622209, 0,
1.52928380839761829735024622209, 2.54678619036393715711638393374, 2.97523965614867243658224775258, 4.86009205886332243916844869170, 4.99055362809910117649143767821, 5.85813553773490752096724667346, 6.40131822132186254897912965187, 7.24606783654665112200275371906, 8.093972413496771858570001723261
