| L(s) = 1 | − 2·2-s + 2·4-s − 5-s − 7-s + 2·10-s + 6·13-s + 2·14-s − 4·16-s − 7·17-s + 5·19-s − 2·20-s + 23-s + 25-s − 12·26-s − 2·28-s − 5·29-s − 8·31-s + 8·32-s + 14·34-s + 35-s − 2·37-s − 10·38-s + 12·41-s + 11·43-s − 2·46-s − 8·47-s + 49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s − 0.377·7-s + 0.632·10-s + 1.66·13-s + 0.534·14-s − 16-s − 1.69·17-s + 1.14·19-s − 0.447·20-s + 0.208·23-s + 1/5·25-s − 2.35·26-s − 0.377·28-s − 0.928·29-s − 1.43·31-s + 1.41·32-s + 2.40·34-s + 0.169·35-s − 0.328·37-s − 1.62·38-s + 1.87·41-s + 1.67·43-s − 0.294·46-s − 1.16·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−15.33029390298316, −14.77404497991088, −13.97957824723868, −13.41801293145894, −13.13116517277136, −12.47836759930534, −11.61658657985945, −11.15965727931200, −10.91648183179516, −10.43867252006878, −9.519032419778141, −9.208386074600587, −8.812217343684658, −8.329932748421153, −7.507764662638401, −7.306864422518988, −6.615231523898964, −5.964969788566979, −5.377026671251394, −4.295740125902604, −3.992048092040066, −3.151480801064470, −2.331801858898330, −1.544130516937588, −0.8413027538816710, 0,
0.8413027538816710, 1.544130516937588, 2.331801858898330, 3.151480801064470, 3.992048092040066, 4.295740125902604, 5.377026671251394, 5.964969788566979, 6.615231523898964, 7.306864422518988, 7.507764662638401, 8.329932748421153, 8.812217343684658, 9.208386074600587, 9.519032419778141, 10.43867252006878, 10.91648183179516, 11.15965727931200, 11.61658657985945, 12.47836759930534, 13.13116517277136, 13.41801293145894, 13.97957824723868, 14.77404497991088, 15.33029390298316
