L(s) = 1 | + 3-s + 2.70·5-s − 7-s + 9-s + 0.701·11-s + 13-s + 2.70·15-s − 2.70·17-s + 0.701·19-s − 21-s − 4.70·23-s + 2.29·25-s + 27-s + 2.70·29-s + 0.701·33-s − 2.70·35-s + 10.7·37-s + 39-s + 3.40·41-s + 10.1·43-s + 2.70·45-s + 8·47-s + 49-s − 2.70·51-s − 2·53-s + 1.89·55-s + 0.701·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.20·5-s − 0.377·7-s + 0.333·9-s + 0.211·11-s + 0.277·13-s + 0.697·15-s − 0.655·17-s + 0.160·19-s − 0.218·21-s − 0.980·23-s + 0.459·25-s + 0.192·27-s + 0.501·29-s + 0.122·33-s − 0.456·35-s + 1.75·37-s + 0.160·39-s + 0.531·41-s + 1.54·43-s + 0.402·45-s + 1.16·47-s + 0.142·49-s − 0.378·51-s − 0.274·53-s + 0.255·55-s + 0.0929·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.072910929\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.072910929\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 2.70T + 5T^{2} \) |
| 11 | \( 1 - 0.701T + 11T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 - 0.701T + 19T^{2} \) |
| 23 | \( 1 + 4.70T + 23T^{2} \) |
| 29 | \( 1 - 2.70T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 3.40T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 - 1.29T + 61T^{2} \) |
| 67 | \( 1 + 5.40T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 1.29T + 73T^{2} \) |
| 79 | \( 1 + 9.40T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 8.80T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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
−8.475082915831101042953912842286, −7.68055848797064308405061146148, −6.85186836284450535489315301650, −6.08981399826918994713517293673, −5.67037557941313795329097237921, −4.49112396178094989313941162990, −3.82542256273838379028773987727, −2.64902807267948658255126067862, −2.17115554342392775260645601725, −0.990966220395811177725472490916,
0.990966220395811177725472490916, 2.17115554342392775260645601725, 2.64902807267948658255126067862, 3.82542256273838379028773987727, 4.49112396178094989313941162990, 5.67037557941313795329097237921, 6.08981399826918994713517293673, 6.85186836284450535489315301650, 7.68055848797064308405061146148, 8.475082915831101042953912842286