| L(s) = 1 | − 0.333·2-s − 0.136·3-s − 1.88·4-s − 5-s + 0.0455·6-s + 1.29·8-s − 2.98·9-s + 0.333·10-s + 5.49·11-s + 0.257·12-s + 0.647·13-s + 0.136·15-s + 3.34·16-s − 0.0895·17-s + 0.995·18-s − 7.20·19-s + 1.88·20-s − 1.83·22-s + 3.01·23-s − 0.176·24-s + 25-s − 0.216·26-s + 0.815·27-s − 0.350·29-s − 0.0455·30-s − 3.40·31-s − 3.71·32-s + ⋯ |
| L(s) = 1 | − 0.236·2-s − 0.0787·3-s − 0.944·4-s − 0.447·5-s + 0.0185·6-s + 0.458·8-s − 0.993·9-s + 0.105·10-s + 1.65·11-s + 0.0743·12-s + 0.179·13-s + 0.0351·15-s + 0.835·16-s − 0.0217·17-s + 0.234·18-s − 1.65·19-s + 0.422·20-s − 0.391·22-s + 0.627·23-s − 0.0361·24-s + 0.200·25-s − 0.0424·26-s + 0.156·27-s − 0.0650·29-s − 0.00830·30-s − 0.612·31-s − 0.656·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 0.333T + 2T^{2} \) |
| 3 | \( 1 + 0.136T + 3T^{2} \) |
| 11 | \( 1 - 5.49T + 11T^{2} \) |
| 13 | \( 1 - 0.647T + 13T^{2} \) |
| 17 | \( 1 + 0.0895T + 17T^{2} \) |
| 19 | \( 1 + 7.20T + 19T^{2} \) |
| 23 | \( 1 - 3.01T + 23T^{2} \) |
| 29 | \( 1 + 0.350T + 29T^{2} \) |
| 31 | \( 1 + 3.40T + 31T^{2} \) |
| 41 | \( 1 + 6.31T + 41T^{2} \) |
| 43 | \( 1 - 4.74T + 43T^{2} \) |
| 47 | \( 1 - 5.52T + 47T^{2} \) |
| 53 | \( 1 + 8.40T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 + 9.27T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 4.13T + 71T^{2} \) |
| 73 | \( 1 + 3.23T + 73T^{2} \) |
| 79 | \( 1 + 5.21T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 + 4.16T + 89T^{2} \) |
| 97 | \( 1 - 3.70T + 97T^{2} \) |
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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
−7.50047104907840483787782699808, −6.60919652915764350495766626306, −6.13110127391480931967617507940, −5.24182092952055719864782700597, −4.51290695394516221460834289913, −3.86420856430919257040653350754, −3.31471113564647138124467101427, −2.06756616432204814802836846698, −1.00545671593733211299686916467, 0,
1.00545671593733211299686916467, 2.06756616432204814802836846698, 3.31471113564647138124467101427, 3.86420856430919257040653350754, 4.51290695394516221460834289913, 5.24182092952055719864782700597, 6.13110127391480931967617507940, 6.60919652915764350495766626306, 7.50047104907840483787782699808
