| L(s) = 1 | − 0.271·2-s − 2.40·3-s − 1.92·4-s + 0.652·6-s − 7-s + 1.06·8-s + 2.78·9-s − 1.85·11-s + 4.63·12-s − 1.69·13-s + 0.271·14-s + 3.56·16-s + 3.27·17-s − 0.755·18-s − 6.98·19-s + 2.40·21-s + 0.503·22-s + 23-s − 2.56·24-s + 0.460·26-s + 0.523·27-s + 1.92·28-s − 1.30·29-s − 3.75·31-s − 3.09·32-s + 4.45·33-s − 0.889·34-s + ⋯ |
| L(s) = 1 | − 0.191·2-s − 1.38·3-s − 0.963·4-s + 0.266·6-s − 0.377·7-s + 0.376·8-s + 0.927·9-s − 0.558·11-s + 1.33·12-s − 0.470·13-s + 0.0725·14-s + 0.890·16-s + 0.794·17-s − 0.178·18-s − 1.60·19-s + 0.524·21-s + 0.107·22-s + 0.208·23-s − 0.523·24-s + 0.0902·26-s + 0.100·27-s + 0.364·28-s − 0.242·29-s − 0.674·31-s − 0.547·32-s + 0.775·33-s − 0.152·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 0.271T + 2T^{2} \) |
| 3 | \( 1 + 2.40T + 3T^{2} \) |
| 11 | \( 1 + 1.85T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 19 | \( 1 + 6.98T + 19T^{2} \) |
| 29 | \( 1 + 1.30T + 29T^{2} \) |
| 31 | \( 1 + 3.75T + 31T^{2} \) |
| 37 | \( 1 - 4.81T + 37T^{2} \) |
| 41 | \( 1 - 0.0960T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 - 8.67T + 53T^{2} \) |
| 59 | \( 1 - 6.86T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 4.91T + 67T^{2} \) |
| 71 | \( 1 + 8.68T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 0.709T + 89T^{2} \) |
| 97 | \( 1 + 9.24T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.123866380170135136586836336364, −7.27860255055677655365657009503, −6.50095824657664626678934691546, −5.65569440275330102147992911122, −5.25252435136533180037520958956, −4.43293923426934017073954253414, −3.68896410925834375510763804019, −2.36543103380641915906680759244, −0.888710752693488787154571605370, 0,
0.888710752693488787154571605370, 2.36543103380641915906680759244, 3.68896410925834375510763804019, 4.43293923426934017073954253414, 5.25252435136533180037520958956, 5.65569440275330102147992911122, 6.50095824657664626678934691546, 7.27860255055677655365657009503, 8.123866380170135136586836336364
