L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 2.82·11-s + 4.82·13-s + 15-s − 3.65·17-s + 2.82·19-s + 21-s + 4·23-s + 25-s + 27-s − 3.65·29-s − 2.82·31-s + 2.82·33-s + 35-s + 0.343·37-s + 4.82·39-s + 3.65·41-s − 9.65·43-s + 45-s + 11.3·47-s + 49-s − 3.65·51-s + 6.48·53-s + 2.82·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.852·11-s + 1.33·13-s + 0.258·15-s − 0.886·17-s + 0.648·19-s + 0.218·21-s + 0.834·23-s + 0.200·25-s + 0.192·27-s − 0.679·29-s − 0.508·31-s + 0.492·33-s + 0.169·35-s + 0.0564·37-s + 0.773·39-s + 0.571·41-s − 1.47·43-s + 0.149·45-s + 1.65·47-s + 0.142·49-s − 0.512·51-s + 0.890·53-s + 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.526345441\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.526345441\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 - 0.828T + 73T^{2} \) |
| 79 | \( 1 + 7.31T + 79T^{2} \) |
| 83 | \( 1 + 6.34T + 83T^{2} \) |
| 89 | \( 1 - 0.343T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.120151440435996492672146225949, −7.17587323374102250790035993067, −6.72130538508751416413037371588, −5.85829997836559913129376573172, −5.20442490003475944838325789943, −4.16754368742790217948686106290, −3.68319029288052364716594794762, −2.71088875382729256610936253916, −1.77458238495113806394634209507, −1.02373242603094179679415200563,
1.02373242603094179679415200563, 1.77458238495113806394634209507, 2.71088875382729256610936253916, 3.68319029288052364716594794762, 4.16754368742790217948686106290, 5.20442490003475944838325789943, 5.85829997836559913129376573172, 6.72130538508751416413037371588, 7.17587323374102250790035993067, 8.120151440435996492672146225949