| L(s) = 1 | + 2-s + 4-s − 2.70·5-s − 7-s + 8-s − 2.70·10-s − 0.701·13-s − 14-s + 16-s + 7.40·19-s − 2.70·20-s + 23-s + 2.29·25-s − 0.701·26-s − 28-s − 6.70·29-s + 6·31-s + 32-s + 2.70·35-s − 2.70·37-s + 7.40·38-s − 2.70·40-s − 1.29·41-s − 0.701·43-s + 46-s + 2.70·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.20·5-s − 0.377·7-s + 0.353·8-s − 0.854·10-s − 0.194·13-s − 0.267·14-s + 0.250·16-s + 1.69·19-s − 0.604·20-s + 0.208·23-s + 0.459·25-s − 0.137·26-s − 0.188·28-s − 1.24·29-s + 1.07·31-s + 0.176·32-s + 0.456·35-s − 0.444·37-s + 1.20·38-s − 0.427·40-s − 0.202·41-s − 0.106·43-s + 0.147·46-s + 0.394·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.124506397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.124506397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2.70T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.701T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 7.40T + 19T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 + 1.29T + 41T^{2} \) |
| 43 | \( 1 + 0.701T + 43T^{2} \) |
| 47 | \( 1 - 2.70T + 47T^{2} \) |
| 53 | \( 1 - 3.40T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 5.40T + 71T^{2} \) |
| 73 | \( 1 - 7.40T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 2.59T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−8.629440024529208531499535649140, −7.77497374755217167412557347728, −7.29231799063202544499763173212, −6.55299686641013405236612339592, −5.50479754903119785543483755890, −4.89685361524396843432395793535, −3.80688298153146717399058336994, −3.45527438155497129447700702379, −2.35857044104519559859925289303, −0.802794875645493798539687683775,
0.802794875645493798539687683775, 2.35857044104519559859925289303, 3.45527438155497129447700702379, 3.80688298153146717399058336994, 4.89685361524396843432395793535, 5.50479754903119785543483755890, 6.55299686641013405236612339592, 7.29231799063202544499763173212, 7.77497374755217167412557347728, 8.629440024529208531499535649140
