| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 11-s − 1.53·13-s + 15-s + 1.75·17-s − 4.82·19-s − 21-s + 3.75·23-s + 25-s − 27-s − 7.86·29-s + 1.53·31-s − 33-s − 35-s + 6.11·37-s + 1.53·39-s + 12.1·41-s + 1.28·43-s − 45-s + 1.53·47-s + 49-s − 1.75·51-s − 8.33·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.301·11-s − 0.425·13-s + 0.258·15-s + 0.425·17-s − 1.10·19-s − 0.218·21-s + 0.783·23-s + 0.200·25-s − 0.192·27-s − 1.46·29-s + 0.275·31-s − 0.174·33-s − 0.169·35-s + 1.00·37-s + 0.245·39-s + 1.89·41-s + 0.196·43-s − 0.149·45-s + 0.223·47-s + 0.142·49-s − 0.245·51-s − 1.14·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.310975624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310975624\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 - 1.75T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 - 1.53T + 31T^{2} \) |
| 37 | \( 1 - 6.11T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 1.28T + 43T^{2} \) |
| 47 | \( 1 - 1.53T + 47T^{2} \) |
| 53 | \( 1 + 8.33T + 53T^{2} \) |
| 59 | \( 1 + 8.35T + 59T^{2} \) |
| 61 | \( 1 + 5.75T + 61T^{2} \) |
| 67 | \( 1 - 3.51T + 67T^{2} \) |
| 71 | \( 1 - 2.95T + 71T^{2} \) |
| 73 | \( 1 - 3.06T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 1.75T + 83T^{2} \) |
| 89 | \( 1 - 3.28T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−8.075859796320594357445491108435, −7.67626470601475619958579650452, −6.84890121138043236840338967527, −6.12879562556333144105399060978, −5.38298626613575605093491915441, −4.53995343537813204295003693691, −3.99903731312605587332948167147, −2.89892546871751708042698527871, −1.82599099578059219593991169718, −0.65984409070085667978747999852,
0.65984409070085667978747999852, 1.82599099578059219593991169718, 2.89892546871751708042698527871, 3.99903731312605587332948167147, 4.53995343537813204295003693691, 5.38298626613575605093491915441, 6.12879562556333144105399060978, 6.84890121138043236840338967527, 7.67626470601475619958579650452, 8.075859796320594357445491108435
