| L(s) = 1 | + 3.22·3-s − 2.37·5-s − 0.397·7-s + 7.42·9-s − 3.21·11-s + 0.155·13-s − 7.66·15-s − 3.28·17-s + 2.84·19-s − 1.28·21-s + 0.630·25-s + 14.2·27-s + 0.378·29-s − 2.74·31-s − 10.3·33-s + 0.944·35-s − 6.42·37-s + 0.501·39-s − 4.00·41-s + 7.43·43-s − 17.6·45-s − 2.97·47-s − 6.84·49-s − 10.5·51-s − 13.0·53-s + 7.61·55-s + 9.18·57-s + ⋯ |
| L(s) = 1 | + 1.86·3-s − 1.06·5-s − 0.150·7-s + 2.47·9-s − 0.968·11-s + 0.0431·13-s − 1.97·15-s − 0.796·17-s + 0.652·19-s − 0.280·21-s + 0.126·25-s + 2.74·27-s + 0.0703·29-s − 0.493·31-s − 1.80·33-s + 0.159·35-s − 1.05·37-s + 0.0803·39-s − 0.625·41-s + 1.13·43-s − 2.62·45-s − 0.433·47-s − 0.977·49-s − 1.48·51-s − 1.79·53-s + 1.02·55-s + 1.21·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3.22T + 3T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 + 0.397T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 13 | \( 1 - 0.155T + 13T^{2} \) |
| 17 | \( 1 + 3.28T + 17T^{2} \) |
| 19 | \( 1 - 2.84T + 19T^{2} \) |
| 29 | \( 1 - 0.378T + 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 + 6.42T + 37T^{2} \) |
| 41 | \( 1 + 4.00T + 41T^{2} \) |
| 43 | \( 1 - 7.43T + 43T^{2} \) |
| 47 | \( 1 + 2.97T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 + 8.64T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 8.40T + 67T^{2} \) |
| 71 | \( 1 + 0.356T + 71T^{2} \) |
| 73 | \( 1 - 5.63T + 73T^{2} \) |
| 79 | \( 1 + 7.53T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 2.16T + 89T^{2} \) |
| 97 | \( 1 - 2.63T + 97T^{2} \) |
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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
−7.59831148053830956772499977871, −7.18341132898164420879737714182, −6.29637227174810968429704894044, −5.02330702719151618037504153526, −4.45503361147431795102068956043, −3.57787180335685266117565295894, −3.21258993748572409187962318761, −2.40633836843702172703854928960, −1.54447487622889567353417106819, 0,
1.54447487622889567353417106819, 2.40633836843702172703854928960, 3.21258993748572409187962318761, 3.57787180335685266117565295894, 4.45503361147431795102068956043, 5.02330702719151618037504153526, 6.29637227174810968429704894044, 7.18341132898164420879737714182, 7.59831148053830956772499977871
