| L(s) = 1 | + 2-s − 2.84·3-s + 4-s − 3.07·5-s − 2.84·6-s + 4.67·7-s + 8-s + 5.11·9-s − 3.07·10-s − 2.84·12-s + 4.99·13-s + 4.67·14-s + 8.74·15-s + 16-s + 3.75·17-s + 5.11·18-s + 19-s − 3.07·20-s − 13.3·21-s − 2.62·23-s − 2.84·24-s + 4.43·25-s + 4.99·26-s − 6.01·27-s + 4.67·28-s + 6.95·29-s + 8.74·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.64·3-s + 0.5·4-s − 1.37·5-s − 1.16·6-s + 1.76·7-s + 0.353·8-s + 1.70·9-s − 0.971·10-s − 0.822·12-s + 1.38·13-s + 1.24·14-s + 2.25·15-s + 0.250·16-s + 0.911·17-s + 1.20·18-s + 0.229·19-s − 0.686·20-s − 2.90·21-s − 0.548·23-s − 0.581·24-s + 0.886·25-s + 0.978·26-s − 1.15·27-s + 0.882·28-s + 1.29·29-s + 1.59·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.906066809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.906066809\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.84T + 3T^{2} \) |
| 5 | \( 1 + 3.07T + 5T^{2} \) |
| 7 | \( 1 - 4.67T + 7T^{2} \) |
| 13 | \( 1 - 4.99T + 13T^{2} \) |
| 17 | \( 1 - 3.75T + 17T^{2} \) |
| 23 | \( 1 + 2.62T + 23T^{2} \) |
| 29 | \( 1 - 6.95T + 29T^{2} \) |
| 31 | \( 1 - 7.18T + 31T^{2} \) |
| 37 | \( 1 + 7.95T + 37T^{2} \) |
| 41 | \( 1 + 9.26T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 - 4.27T + 47T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 - 0.750T + 67T^{2} \) |
| 71 | \( 1 - 2.29T + 71T^{2} \) |
| 73 | \( 1 + 9.81T + 73T^{2} \) |
| 79 | \( 1 - 4.49T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 - 2.22T + 89T^{2} \) |
| 97 | \( 1 - 4.25T + 97T^{2} \) |
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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
−8.123955023654008471328762926184, −7.51032417480362115810045999025, −6.74135495583243634236413845007, −5.92594408242641704346441624161, −5.31482960968466787888746447817, −4.58160888172531723790269227204, −4.20423639714106230474368898938, −3.24628198373222228075746808587, −1.57130222732879845028991938761, −0.835394871678958054893572637408,
0.835394871678958054893572637408, 1.57130222732879845028991938761, 3.24628198373222228075746808587, 4.20423639714106230474368898938, 4.58160888172531723790269227204, 5.31482960968466787888746447817, 5.92594408242641704346441624161, 6.74135495583243634236413845007, 7.51032417480362115810045999025, 8.123955023654008471328762926184
