L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 6·11-s + 12-s + 13-s + 16-s + 2·17-s − 18-s − 19-s + 6·22-s − 23-s − 24-s − 26-s + 27-s − 9·29-s − 2·31-s − 32-s − 6·33-s − 2·34-s + 36-s + 10·37-s + 38-s + 39-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s + 1.27·22-s − 0.208·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s − 1.67·29-s − 0.359·31-s − 0.176·32-s − 1.04·33-s − 0.342·34-s + 1/6·36-s + 1.64·37-s + 0.162·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.55805867609173, −13.10020362469025, −12.80834432769950, −12.31545613070480, −11.58990923275808, −11.06492511481176, −10.77213572333029, −10.16959238435254, −9.803898487531253, −9.314769019961112, −8.771478117158685, −8.243098955995817, −7.848671645926810, −7.423097729734987, −7.098335431944313, −6.147768130352053, −5.780033638596675, −5.267517374121834, −4.573508497799537, −3.905679365612714, −3.308452261715565, −2.683248803957509, −2.264729833269847, −1.628771979883575, −0.7624753647897921, 0,
0.7624753647897921, 1.628771979883575, 2.264729833269847, 2.683248803957509, 3.308452261715565, 3.905679365612714, 4.573508497799537, 5.267517374121834, 5.780033638596675, 6.147768130352053, 7.098335431944313, 7.423097729734987, 7.848671645926810, 8.243098955995817, 8.771478117158685, 9.314769019961112, 9.803898487531253, 10.16959238435254, 10.77213572333029, 11.06492511481176, 11.58990923275808, 12.31545613070480, 12.80834432769950, 13.10020362469025, 13.55805867609173