L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 4·11-s − 3·13-s + 14-s + 16-s + 7·17-s − 6·19-s + 4·22-s + 9·23-s + 3·26-s − 28-s − 3·29-s − 7·31-s − 32-s − 7·34-s − 10·37-s + 6·38-s + 41-s + 13·43-s − 4·44-s − 9·46-s − 2·47-s + 49-s − 3·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.20·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 1.69·17-s − 1.37·19-s + 0.852·22-s + 1.87·23-s + 0.588·26-s − 0.188·28-s − 0.557·29-s − 1.25·31-s − 0.176·32-s − 1.20·34-s − 1.64·37-s + 0.973·38-s + 0.156·41-s + 1.98·43-s − 0.603·44-s − 1.32·46-s − 0.291·47-s + 1/7·49-s − 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9195732093\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9195732093\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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
−8.781675080601501623768376748522, −7.84335343978744776618511390271, −7.36845273584414996909558838079, −6.66264157947553488246671032551, −5.51657674960247696910072397970, −5.14823767607295882644190048281, −3.77034495441787172485473312012, −2.88780756149487526860144751240, −2.05198583129271445231774373106, −0.62005820358442542220858327088,
0.62005820358442542220858327088, 2.05198583129271445231774373106, 2.88780756149487526860144751240, 3.77034495441787172485473312012, 5.14823767607295882644190048281, 5.51657674960247696910072397970, 6.66264157947553488246671032551, 7.36845273584414996909558838079, 7.84335343978744776618511390271, 8.781675080601501623768376748522