| L(s) = 1 | − 2.43·2-s + 2.82·3-s + 3.92·4-s − 6.87·6-s + 7-s − 4.69·8-s + 4.97·9-s − 2.34·11-s + 11.0·12-s − 2.65·13-s − 2.43·14-s + 3.56·16-s − 0.598·17-s − 12.1·18-s − 4.73·19-s + 2.82·21-s + 5.71·22-s − 23-s − 13.2·24-s + 6.46·26-s + 5.59·27-s + 3.92·28-s + 3.31·29-s − 7.99·31-s + 0.697·32-s − 6.63·33-s + 1.45·34-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.63·3-s + 1.96·4-s − 2.80·6-s + 0.377·7-s − 1.65·8-s + 1.65·9-s − 0.707·11-s + 3.20·12-s − 0.736·13-s − 0.650·14-s + 0.891·16-s − 0.145·17-s − 2.85·18-s − 1.08·19-s + 0.616·21-s + 1.21·22-s − 0.208·23-s − 2.70·24-s + 1.26·26-s + 1.07·27-s + 0.742·28-s + 0.616·29-s − 1.43·31-s + 0.123·32-s − 1.15·33-s + 0.250·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 3 | \( 1 - 2.82T + 3T^{2} \) |
| 11 | \( 1 + 2.34T + 11T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 + 0.598T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 29 | \( 1 - 3.31T + 29T^{2} \) |
| 31 | \( 1 + 7.99T + 31T^{2} \) |
| 37 | \( 1 + 8.51T + 37T^{2} \) |
| 41 | \( 1 + 4.06T + 41T^{2} \) |
| 43 | \( 1 - 8.27T + 43T^{2} \) |
| 47 | \( 1 - 1.44T + 47T^{2} \) |
| 53 | \( 1 - 1.87T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 7.24T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 4.03T + 83T^{2} \) |
| 89 | \( 1 + 5.22T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−8.321906940223024267335019341239, −7.50444447516487424499034573117, −7.32817674835484554193880977131, −6.29300307548556933419698300734, −5.01430579231561473542973054871, −3.98122795961882221598555187952, −2.89386641588467549071271253194, −2.23816003396325931656421028615, −1.59928535440952440252382862189, 0,
1.59928535440952440252382862189, 2.23816003396325931656421028615, 2.89386641588467549071271253194, 3.98122795961882221598555187952, 5.01430579231561473542973054871, 6.29300307548556933419698300734, 7.32817674835484554193880977131, 7.50444447516487424499034573117, 8.321906940223024267335019341239
