L(s) = 1 | − 3-s + 3.20·5-s − 2.04·7-s + 9-s + 0.355·11-s − 1.61·13-s − 3.20·15-s − 1.55·17-s + 4.81·19-s + 2.04·21-s − 1.33·23-s + 5.28·25-s − 27-s + 7.93·29-s + 0.483·31-s − 0.355·33-s − 6.55·35-s + 1.13·37-s + 1.61·39-s + 2.30·41-s − 8.90·43-s + 3.20·45-s − 8.25·47-s − 2.81·49-s + 1.55·51-s + 5.66·53-s + 1.13·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.43·5-s − 0.773·7-s + 0.333·9-s + 0.107·11-s − 0.447·13-s − 0.827·15-s − 0.376·17-s + 1.10·19-s + 0.446·21-s − 0.278·23-s + 1.05·25-s − 0.192·27-s + 1.47·29-s + 0.0868·31-s − 0.0618·33-s − 1.10·35-s + 0.187·37-s + 0.258·39-s + 0.360·41-s − 1.35·43-s + 0.478·45-s − 1.20·47-s − 0.402·49-s + 0.217·51-s + 0.777·53-s + 0.153·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.923327071\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.923327071\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
good | 5 | \( 1 - 3.20T + 5T^{2} \) |
| 7 | \( 1 + 2.04T + 7T^{2} \) |
| 11 | \( 1 - 0.355T + 11T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 + 1.55T + 17T^{2} \) |
| 19 | \( 1 - 4.81T + 19T^{2} \) |
| 23 | \( 1 + 1.33T + 23T^{2} \) |
| 29 | \( 1 - 7.93T + 29T^{2} \) |
| 31 | \( 1 - 0.483T + 31T^{2} \) |
| 37 | \( 1 - 1.13T + 37T^{2} \) |
| 41 | \( 1 - 2.30T + 41T^{2} \) |
| 43 | \( 1 + 8.90T + 43T^{2} \) |
| 47 | \( 1 + 8.25T + 47T^{2} \) |
| 53 | \( 1 - 5.66T + 53T^{2} \) |
| 59 | \( 1 - 8.23T + 59T^{2} \) |
| 61 | \( 1 - 9.44T + 61T^{2} \) |
| 67 | \( 1 - 3.12T + 67T^{2} \) |
| 71 | \( 1 - 2.32T + 71T^{2} \) |
| 73 | \( 1 + 5.59T + 73T^{2} \) |
| 79 | \( 1 - 1.77T + 79T^{2} \) |
| 83 | \( 1 - 1.56T + 83T^{2} \) |
| 89 | \( 1 - 8.47T + 89T^{2} \) |
| 97 | \( 1 - 19.3T + 97T^{2} \) |
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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
−8.092460433262617273780269283965, −7.05680281853266114763943235327, −6.55446401561524259072281948899, −6.01798551668109598764121604966, −5.26601877203548109774198129641, −4.73261844009382130668516395186, −3.54722211793180244273023232096, −2.69136299628662306498366466685, −1.83929932821646630391721846243, −0.75485184158611352864356976433,
0.75485184158611352864356976433, 1.83929932821646630391721846243, 2.69136299628662306498366466685, 3.54722211793180244273023232096, 4.73261844009382130668516395186, 5.26601877203548109774198129641, 6.01798551668109598764121604966, 6.55446401561524259072281948899, 7.05680281853266114763943235327, 8.092460433262617273780269283965