| L(s) = 1 | − 3-s − 2·4-s + 5-s + 9-s + 4·11-s + 2·12-s + 5·13-s − 15-s + 4·16-s − 7·17-s − 2·20-s + 25-s − 27-s − 5·29-s − 2·31-s − 4·33-s − 2·36-s + 2·37-s − 5·39-s − 4·41-s − 10·43-s − 8·44-s + 45-s + 3·47-s − 4·48-s + 7·51-s − 10·52-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.577·12-s + 1.38·13-s − 0.258·15-s + 16-s − 1.69·17-s − 0.447·20-s + 1/5·25-s − 0.192·27-s − 0.928·29-s − 0.359·31-s − 0.696·33-s − 1/3·36-s + 0.328·37-s − 0.800·39-s − 0.624·41-s − 1.52·43-s − 1.20·44-s + 0.149·45-s + 0.437·47-s − 0.577·48-s + 0.980·51-s − 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.298747372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.298747372\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 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
−12.55695603720060, −12.01317560509474, −11.45660406045085, −11.08427589441568, −10.79101706649402, −10.19666555351046, −9.574252292065197, −9.358236493845103, −8.848444151249348, −8.527306841475919, −8.092517707503355, −7.282769730414743, −6.815492636616567, −6.402337377527376, −5.897980920646621, −5.642851997268468, −4.849650669614785, −4.488518876133335, −4.090847722028585, −3.487986294449595, −3.135389989316335, −1.959688040461649, −1.703012671795624, −1.036231007562833, −0.3468925487212402,
0.3468925487212402, 1.036231007562833, 1.703012671795624, 1.959688040461649, 3.135389989316335, 3.487986294449595, 4.090847722028585, 4.488518876133335, 4.849650669614785, 5.642851997268468, 5.897980920646621, 6.402337377527376, 6.815492636616567, 7.282769730414743, 8.092517707503355, 8.527306841475919, 8.848444151249348, 9.358236493845103, 9.574252292065197, 10.19666555351046, 10.79101706649402, 11.08427589441568, 11.45660406045085, 12.01317560509474, 12.55695603720060
