| L(s) = 1 | − 2·2-s + 4·4-s − 19·7-s − 8·8-s − 12·11-s + 50·13-s + 38·14-s + 16·16-s + 126·17-s + 29·19-s + 24·22-s − 18·23-s − 100·26-s − 76·28-s − 102·29-s − 265·31-s − 32·32-s − 252·34-s + 65·37-s − 58·38-s − 240·41-s − 367·43-s − 48·44-s + 36·46-s + 72·47-s + 18·49-s + 200·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.02·7-s − 0.353·8-s − 0.328·11-s + 1.06·13-s + 0.725·14-s + 1/4·16-s + 1.79·17-s + 0.350·19-s + 0.232·22-s − 0.163·23-s − 0.754·26-s − 0.512·28-s − 0.653·29-s − 1.53·31-s − 0.176·32-s − 1.27·34-s + 0.288·37-s − 0.247·38-s − 0.914·41-s − 1.30·43-s − 0.164·44-s + 0.115·46-s + 0.223·47-s + 0.0524·49-s + 0.533·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.197756965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.197756965\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 19 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 126 T + p^{3} T^{2} \) |
| 19 | \( 1 - 29 T + p^{3} T^{2} \) |
| 23 | \( 1 + 18 T + p^{3} T^{2} \) |
| 29 | \( 1 + 102 T + p^{3} T^{2} \) |
| 31 | \( 1 + 265 T + p^{3} T^{2} \) |
| 37 | \( 1 - 65 T + p^{3} T^{2} \) |
| 41 | \( 1 + 240 T + p^{3} T^{2} \) |
| 43 | \( 1 + 367 T + p^{3} T^{2} \) |
| 47 | \( 1 - 72 T + p^{3} T^{2} \) |
| 53 | \( 1 - 12 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 102 T + p^{3} T^{2} \) |
| 61 | \( 1 + 103 T + p^{3} T^{2} \) |
| 67 | \( 1 + 52 T + p^{3} T^{2} \) |
| 71 | \( 1 - 582 T + p^{3} T^{2} \) |
| 73 | \( 1 - 65 T + p^{3} T^{2} \) |
| 79 | \( 1 - 173 T + p^{3} T^{2} \) |
| 83 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 89 | \( 1 - 822 T + p^{3} T^{2} \) |
| 97 | \( 1 - 821 T + p^{3} T^{2} \) |
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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
−9.318900538686485615576426789186, −8.475399727253019003304133822603, −7.67536801077577485583991258444, −6.91575265726053695466035987088, −5.97765310895290684446565812420, −5.35062664406461979960493018996, −3.67227248628184297493738513575, −3.18616111031305912183932266030, −1.76188975576935124138012702890, −0.61320999322795966958727272036,
0.61320999322795966958727272036, 1.76188975576935124138012702890, 3.18616111031305912183932266030, 3.67227248628184297493738513575, 5.35062664406461979960493018996, 5.97765310895290684446565812420, 6.91575265726053695466035987088, 7.67536801077577485583991258444, 8.475399727253019003304133822603, 9.318900538686485615576426789186
