| L(s) = 1 | + 2.75·3-s − 3.14·5-s − 3.24·7-s + 4.59·9-s + 6.57·11-s − 0.801·13-s − 8.66·15-s − 4.08·17-s − 4.28·19-s − 8.95·21-s + 4.89·25-s + 4.39·27-s − 0.891·29-s + 0.691·31-s + 18.1·33-s + 10.2·35-s + 0.676·37-s − 2.20·39-s + 4.75·41-s + 3.07·43-s − 14.4·45-s − 8.85·47-s + 3.55·49-s − 11.2·51-s − 10.8·53-s − 20.6·55-s − 11.8·57-s + ⋯ |
| L(s) = 1 | + 1.59·3-s − 1.40·5-s − 1.22·7-s + 1.53·9-s + 1.98·11-s − 0.222·13-s − 2.23·15-s − 0.990·17-s − 0.984·19-s − 1.95·21-s + 0.978·25-s + 0.845·27-s − 0.165·29-s + 0.124·31-s + 3.15·33-s + 1.72·35-s + 0.111·37-s − 0.353·39-s + 0.742·41-s + 0.468·43-s − 2.15·45-s − 1.29·47-s + 0.507·49-s − 1.57·51-s − 1.49·53-s − 2.78·55-s − 1.56·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.75T + 3T^{2} \) |
| 5 | \( 1 + 3.14T + 5T^{2} \) |
| 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 - 6.57T + 11T^{2} \) |
| 13 | \( 1 + 0.801T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 19 | \( 1 + 4.28T + 19T^{2} \) |
| 29 | \( 1 + 0.891T + 29T^{2} \) |
| 31 | \( 1 - 0.691T + 31T^{2} \) |
| 37 | \( 1 - 0.676T + 37T^{2} \) |
| 41 | \( 1 - 4.75T + 41T^{2} \) |
| 43 | \( 1 - 3.07T + 43T^{2} \) |
| 47 | \( 1 + 8.85T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 5.57T + 59T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 67 | \( 1 + 4.45T + 67T^{2} \) |
| 71 | \( 1 + 3.47T + 71T^{2} \) |
| 73 | \( 1 + 3.07T + 73T^{2} \) |
| 79 | \( 1 - 0.667T + 79T^{2} \) |
| 83 | \( 1 + 3.33T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 9.14T + 97T^{2} \) |
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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
−8.228700477020309208066013526769, −7.34746419902789026761033557370, −6.76682421149552019673554265364, −6.21238618408217046598303139017, −4.37680844029334603813919941795, −4.11421851073971205473344515547, −3.41012671064845600216616128652, −2.75655957931777682169706574228, −1.57815875676999702689557542755, 0,
1.57815875676999702689557542755, 2.75655957931777682169706574228, 3.41012671064845600216616128652, 4.11421851073971205473344515547, 4.37680844029334603813919941795, 6.21238618408217046598303139017, 6.76682421149552019673554265364, 7.34746419902789026761033557370, 8.228700477020309208066013526769
