| L(s) = 1 | − 0.782·3-s − 2.11·5-s − 1.82·7-s − 2.38·9-s − 4.89·11-s − 2.22·13-s + 1.65·15-s − 6.25·17-s − 6.38·19-s + 1.42·21-s + 4.73·23-s − 0.512·25-s + 4.21·27-s − 6.43·29-s + 7.58·31-s + 3.82·33-s + 3.87·35-s + 2.60·37-s + 1.73·39-s + 10.1·41-s − 43-s + 5.05·45-s − 3.16·47-s − 3.66·49-s + 4.89·51-s − 0.336·53-s + 10.3·55-s + ⋯ |
| L(s) = 1 | − 0.451·3-s − 0.947·5-s − 0.690·7-s − 0.796·9-s − 1.47·11-s − 0.615·13-s + 0.427·15-s − 1.51·17-s − 1.46·19-s + 0.311·21-s + 0.986·23-s − 0.102·25-s + 0.811·27-s − 1.19·29-s + 1.36·31-s + 0.666·33-s + 0.654·35-s + 0.428·37-s + 0.278·39-s + 1.58·41-s − 0.152·43-s + 0.754·45-s − 0.461·47-s − 0.523·49-s + 0.685·51-s − 0.0462·53-s + 1.39·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1600948993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1600948993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 0.782T + 3T^{2} \) |
| 5 | \( 1 + 2.11T + 5T^{2} \) |
| 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 2.22T + 13T^{2} \) |
| 17 | \( 1 + 6.25T + 17T^{2} \) |
| 19 | \( 1 + 6.38T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 + 6.43T + 29T^{2} \) |
| 31 | \( 1 - 7.58T + 31T^{2} \) |
| 37 | \( 1 - 2.60T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 47 | \( 1 + 3.16T + 47T^{2} \) |
| 53 | \( 1 + 0.336T + 53T^{2} \) |
| 59 | \( 1 - 2.21T + 59T^{2} \) |
| 61 | \( 1 + 3.28T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 17.4T + 79T^{2} \) |
| 83 | \( 1 - 7.45T + 83T^{2} \) |
| 89 | \( 1 + 0.0458T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.694248012637290645261891386834, −8.081119000018173242879716060184, −7.31776978754972868159070886817, −6.50816065739840641030858192158, −5.80015706344057206531409896971, −4.81857995160323611331121601459, −4.22389372507016062440525356743, −2.98496378895922539218058945433, −2.38488566261009267077400925418, −0.23181704887328679369971800253,
0.23181704887328679369971800253, 2.38488566261009267077400925418, 2.98496378895922539218058945433, 4.22389372507016062440525356743, 4.81857995160323611331121601459, 5.80015706344057206531409896971, 6.50816065739840641030858192158, 7.31776978754972868159070886817, 8.081119000018173242879716060184, 8.694248012637290645261891386834
