| L(s) = 1 | + 2.56·2-s − 0.961·3-s + 4.60·4-s − 2.47·6-s − 2.40·7-s + 6.69·8-s − 2.07·9-s + 2.76·11-s − 4.42·12-s − 1.28·13-s − 6.19·14-s + 7.98·16-s − 6.01·17-s − 5.33·18-s + 2.42·19-s + 2.31·21-s + 7.09·22-s − 4.77·23-s − 6.43·24-s − 3.29·26-s + 4.88·27-s − 11.0·28-s + 7.77·29-s − 4.82·31-s + 7.14·32-s − 2.65·33-s − 15.4·34-s + ⋯ |
| L(s) = 1 | + 1.81·2-s − 0.555·3-s + 2.30·4-s − 1.00·6-s − 0.910·7-s + 2.36·8-s − 0.691·9-s + 0.832·11-s − 1.27·12-s − 0.355·13-s − 1.65·14-s + 1.99·16-s − 1.45·17-s − 1.25·18-s + 0.556·19-s + 0.505·21-s + 1.51·22-s − 0.996·23-s − 1.31·24-s − 0.646·26-s + 0.939·27-s − 2.09·28-s + 1.44·29-s − 0.865·31-s + 1.26·32-s − 0.462·33-s − 2.64·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 3 | \( 1 + 0.961T + 3T^{2} \) |
| 7 | \( 1 + 2.40T + 7T^{2} \) |
| 11 | \( 1 - 2.76T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 + 6.01T + 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 23 | \( 1 + 4.77T + 23T^{2} \) |
| 29 | \( 1 - 7.77T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 + 9.25T + 37T^{2} \) |
| 41 | \( 1 - 3.19T + 41T^{2} \) |
| 43 | \( 1 + 7.89T + 43T^{2} \) |
| 47 | \( 1 + 4.18T + 47T^{2} \) |
| 53 | \( 1 + 2.01T + 53T^{2} \) |
| 59 | \( 1 - 7.91T + 59T^{2} \) |
| 61 | \( 1 + 6.64T + 61T^{2} \) |
| 67 | \( 1 + 1.72T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 8.98T + 73T^{2} \) |
| 79 | \( 1 + 3.90T + 79T^{2} \) |
| 83 | \( 1 - 3.72T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 4.15T + 97T^{2} \) |
| 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
−7.16214917166083917527441744194, −6.59763695804254980197025840062, −6.27650761231413701439627746249, −5.50778001342229084029966813972, −4.85207697007647319969787246239, −4.13280833334270644674436704187, −3.36438249043567970861942496832, −2.72388904714551310272326672305, −1.73634483780002965928949689301, 0,
1.73634483780002965928949689301, 2.72388904714551310272326672305, 3.36438249043567970861942496832, 4.13280833334270644674436704187, 4.85207697007647319969787246239, 5.50778001342229084029966813972, 6.27650761231413701439627746249, 6.59763695804254980197025840062, 7.16214917166083917527441744194
