| L(s) = 1 | + 3-s − 0.716·5-s − 3.33·7-s + 9-s − 1.47·11-s − 1.07·13-s − 0.716·15-s + 4.68·17-s − 1.60·19-s − 3.33·21-s + 5.96·23-s − 4.48·25-s + 27-s + 4.73·29-s + 8.82·31-s − 1.47·33-s + 2.38·35-s − 10.4·37-s − 1.07·39-s − 2.92·41-s − 8.38·43-s − 0.716·45-s + 6.24·47-s + 4.08·49-s + 4.68·51-s − 9.50·53-s + 1.05·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.320·5-s − 1.25·7-s + 0.333·9-s − 0.443·11-s − 0.297·13-s − 0.184·15-s + 1.13·17-s − 0.367·19-s − 0.726·21-s + 1.24·23-s − 0.897·25-s + 0.192·27-s + 0.878·29-s + 1.58·31-s − 0.256·33-s + 0.403·35-s − 1.71·37-s − 0.171·39-s − 0.456·41-s − 1.27·43-s − 0.106·45-s + 0.910·47-s + 0.584·49-s + 0.656·51-s − 1.30·53-s + 0.142·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.682102820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.682102820\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 0.716T + 5T^{2} \) |
| 7 | \( 1 + 3.33T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 + 1.07T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 23 | \( 1 - 5.96T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 - 8.82T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 2.92T + 41T^{2} \) |
| 43 | \( 1 + 8.38T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 + 9.50T + 53T^{2} \) |
| 59 | \( 1 + 4.74T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 7.30T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 1.24T + 83T^{2} \) |
| 89 | \( 1 + 7.67T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.337876787299501561504173275636, −7.86743269956584880668784063275, −6.88375913224511693908078860284, −6.50475893188823352760808405089, −5.40399836750024361719032491916, −4.67015891328335782807944683512, −3.46472421167404784525848669852, −3.20823190650998150418392019065, −2.15571078608387514564322932728, −0.70454365069032092724161603499,
0.70454365069032092724161603499, 2.15571078608387514564322932728, 3.20823190650998150418392019065, 3.46472421167404784525848669852, 4.67015891328335782807944683512, 5.40399836750024361719032491916, 6.50475893188823352760808405089, 6.88375913224511693908078860284, 7.86743269956584880668784063275, 8.337876787299501561504173275636
