L(s) = 1 | + 0.818·3-s − 0.570·5-s + 7-s − 2.33·9-s + 11-s + 13-s − 0.466·15-s − 6.67·17-s + 1.44·19-s + 0.818·21-s + 5.53·23-s − 4.67·25-s − 4.36·27-s − 5.42·29-s − 1.22·31-s + 0.818·33-s − 0.570·35-s − 4.75·37-s + 0.818·39-s + 3.95·41-s + 8.80·43-s + 1.32·45-s + 7.01·47-s + 49-s − 5.45·51-s + 0.903·53-s − 0.570·55-s + ⋯ |
L(s) = 1 | + 0.472·3-s − 0.254·5-s + 0.377·7-s − 0.776·9-s + 0.301·11-s + 0.277·13-s − 0.120·15-s − 1.61·17-s + 0.331·19-s + 0.178·21-s + 1.15·23-s − 0.935·25-s − 0.839·27-s − 1.00·29-s − 0.220·31-s + 0.142·33-s − 0.0963·35-s − 0.781·37-s + 0.130·39-s + 0.618·41-s + 1.34·43-s + 0.198·45-s + 1.02·47-s + 0.142·49-s − 0.764·51-s + 0.124·53-s − 0.0768·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.952174382\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.952174382\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 0.818T + 3T^{2} \) |
| 5 | \( 1 + 0.570T + 5T^{2} \) |
| 17 | \( 1 + 6.67T + 17T^{2} \) |
| 19 | \( 1 - 1.44T + 19T^{2} \) |
| 23 | \( 1 - 5.53T + 23T^{2} \) |
| 29 | \( 1 + 5.42T + 29T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 + 4.75T + 37T^{2} \) |
| 41 | \( 1 - 3.95T + 41T^{2} \) |
| 43 | \( 1 - 8.80T + 43T^{2} \) |
| 47 | \( 1 - 7.01T + 47T^{2} \) |
| 53 | \( 1 - 0.903T + 53T^{2} \) |
| 59 | \( 1 - 7.67T + 59T^{2} \) |
| 61 | \( 1 - 4.80T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 2.78T + 71T^{2} \) |
| 73 | \( 1 - 0.712T + 73T^{2} \) |
| 79 | \( 1 - 7.71T + 79T^{2} \) |
| 83 | \( 1 - 8.30T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 1.86T + 97T^{2} \) |
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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
−7.86681339476542044663266346186, −7.23382109303960012326723319146, −6.53031414634725722079041931553, −5.68829995512517993167979730726, −5.08950151244398265309000207242, −4.08493238490994994130590948810, −3.63093082997509771409319203843, −2.56540961288140907432160260832, −1.98371449543261092895234233216, −0.65702969983056650530479871671,
0.65702969983056650530479871671, 1.98371449543261092895234233216, 2.56540961288140907432160260832, 3.63093082997509771409319203843, 4.08493238490994994130590948810, 5.08950151244398265309000207242, 5.68829995512517993167979730726, 6.53031414634725722079041931553, 7.23382109303960012326723319146, 7.86681339476542044663266346186