| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 4·7-s − 8-s + 9-s + 10-s + 12-s + 4·14-s − 15-s + 16-s + 6·17-s − 18-s + 19-s − 20-s − 4·21-s + 3·23-s − 24-s + 25-s + 27-s − 4·28-s + 4·29-s + 30-s + 2·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.06·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.872·21-s + 0.625·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.755·28-s + 0.742·29-s + 0.182·30-s + 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.36489261809024, −13.18402175308969, −12.40557802497577, −12.18517660070608, −11.78330393305550, −10.98047177473380, −10.56243021560488, −10.02493949675716, −9.617515525148172, −9.411585843266161, −8.607118645525632, −8.394009493401297, −7.706624256319954, −7.260198991424253, −6.917368193532140, −6.206796795638278, −5.906535641481395, −5.105254364039686, −4.482311145213781, −3.649585880741393, −3.342167716161912, −2.878437383892370, −2.338419550709967, −1.319680651778313, −0.8357985087230142, 0,
0.8357985087230142, 1.319680651778313, 2.338419550709967, 2.878437383892370, 3.342167716161912, 3.649585880741393, 4.482311145213781, 5.105254364039686, 5.906535641481395, 6.206796795638278, 6.917368193532140, 7.260198991424253, 7.706624256319954, 8.394009493401297, 8.607118645525632, 9.411585843266161, 9.617515525148172, 10.02493949675716, 10.56243021560488, 10.98047177473380, 11.78330393305550, 12.18517660070608, 12.40557802497577, 13.18402175308969, 13.36489261809024
