| L(s) = 1 | + 2.23·3-s − 2.28·5-s + 3.70·7-s + 2.00·9-s + 0.540·11-s − 6.23·13-s − 5.11·15-s − 3.70·17-s + 0.333·19-s + 8.27·21-s + 0.236·25-s − 2.23·27-s − 8.70·29-s − 7.94·31-s + 1.20·33-s − 8.47·35-s + 9.69·37-s − 13.9·39-s + 2.70·41-s − 8.48·43-s − 4.57·45-s − 9.47·47-s + 6.70·49-s − 8.27·51-s + 6.65·53-s − 1.23·55-s + 0.746·57-s + ⋯ |
| L(s) = 1 | + 1.29·3-s − 1.02·5-s + 1.39·7-s + 0.666·9-s + 0.162·11-s − 1.72·13-s − 1.32·15-s − 0.897·17-s + 0.0765·19-s + 1.80·21-s + 0.0472·25-s − 0.430·27-s − 1.61·29-s − 1.42·31-s + 0.210·33-s − 1.43·35-s + 1.59·37-s − 2.23·39-s + 0.422·41-s − 1.29·43-s − 0.682·45-s − 1.38·47-s + 0.958·49-s − 1.15·51-s + 0.914·53-s − 0.166·55-s + 0.0988·57-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 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 7 | \( 1 - 3.70T + 7T^{2} \) |
| 11 | \( 1 - 0.540T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 - 0.333T + 19T^{2} \) |
| 29 | \( 1 + 8.70T + 29T^{2} \) |
| 31 | \( 1 + 7.94T + 31T^{2} \) |
| 37 | \( 1 - 9.69T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 9.47T + 47T^{2} \) |
| 53 | \( 1 - 6.65T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 9.48T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 - 3.76T + 71T^{2} \) |
| 73 | \( 1 + 8.23T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 7.19T + 83T^{2} \) |
| 89 | \( 1 + 3.90T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.997494096223462751568821935676, −7.49861611653573836405605094850, −7.10006815943775758467596413460, −5.64435950939342625950363888636, −4.76157498721376990398376677166, −4.20021196318260005730156451080, −3.40923305384672370564399850570, −2.37369517441538133818800001219, −1.79008011979486974669513582231, 0,
1.79008011979486974669513582231, 2.37369517441538133818800001219, 3.40923305384672370564399850570, 4.20021196318260005730156451080, 4.76157498721376990398376677166, 5.64435950939342625950363888636, 7.10006815943775758467596413460, 7.49861611653573836405605094850, 7.997494096223462751568821935676
