| L(s) = 1 | − 2-s + 3-s + 4-s − 4·5-s − 6-s − 7-s − 8-s + 9-s + 4·10-s + 3·11-s + 12-s − 4·13-s + 14-s − 4·15-s + 16-s + 2·17-s − 18-s − 4·20-s − 21-s − 3·22-s + 23-s − 24-s + 11·25-s + 4·26-s + 27-s − 28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.904·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s − 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.894·20-s − 0.218·21-s − 0.639·22-s + 0.208·23-s − 0.204·24-s + 11/5·25-s + 0.784·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.68671742167836, −12.45117779967901, −11.99557522667871, −11.72549605968372, −11.28802303803644, −10.68298302016593, −10.13385112807975, −9.901130731620603, −9.243056225035915, −8.801725730893809, −8.467805241678862, −7.933689823662557, −7.649266752849892, −7.051296034820223, −6.804434225120758, −6.373448553058961, −5.450820665963632, −4.893181179693390, −4.347700282362915, −3.944084805614686, −3.296758600218543, −2.998699556907231, −2.475781383983170, −1.457089840514886, −1.143344439650934, 0, 0,
1.143344439650934, 1.457089840514886, 2.475781383983170, 2.998699556907231, 3.296758600218543, 3.944084805614686, 4.347700282362915, 4.893181179693390, 5.450820665963632, 6.373448553058961, 6.804434225120758, 7.051296034820223, 7.649266752849892, 7.933689823662557, 8.467805241678862, 8.801725730893809, 9.243056225035915, 9.901130731620603, 10.13385112807975, 10.68298302016593, 11.28802303803644, 11.72549605968372, 11.99557522667871, 12.45117779967901, 12.68671742167836
