L(s) = 1 | − 1.56·3-s − 5-s − 7-s − 0.561·9-s + 5.56·11-s − 3.56·13-s + 1.56·15-s − 6.68·17-s + 4·19-s + 1.56·21-s + 25-s + 5.56·27-s + 0.438·29-s − 3.12·31-s − 8.68·33-s + 35-s + 9.12·37-s + 5.56·39-s + 8.24·41-s + 0.561·45-s + 2.43·47-s + 49-s + 10.4·51-s + 1.12·53-s − 5.56·55-s − 6.24·57-s − 2.24·59-s + ⋯ |
L(s) = 1 | − 0.901·3-s − 0.447·5-s − 0.377·7-s − 0.187·9-s + 1.67·11-s − 0.987·13-s + 0.403·15-s − 1.62·17-s + 0.917·19-s + 0.340·21-s + 0.200·25-s + 1.07·27-s + 0.0814·29-s − 0.560·31-s − 1.51·33-s + 0.169·35-s + 1.49·37-s + 0.890·39-s + 1.28·41-s + 0.0837·45-s + 0.355·47-s + 0.142·49-s + 1.46·51-s + 0.154·53-s − 0.749·55-s − 0.827·57-s − 0.292·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8868241339\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8868241339\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 0.438T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 - 9.12T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2.43T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 - 8.24T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 + 5.56T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 0.246T + 89T^{2} \) |
| 97 | \( 1 + 6.68T + 97T^{2} \) |
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
−9.636846512157124384757463331095, −9.250686732923454744447362339835, −8.195253292694092301854847386123, −7.02508163233220331888450504605, −6.57533058423456775307880248290, −5.63818107383750583188857847540, −4.62872599688903493914376376321, −3.81976176145613236921219757541, −2.46919846098296919657998920612, −0.73721805423528470350407909024,
0.73721805423528470350407909024, 2.46919846098296919657998920612, 3.81976176145613236921219757541, 4.62872599688903493914376376321, 5.63818107383750583188857847540, 6.57533058423456775307880248290, 7.02508163233220331888450504605, 8.195253292694092301854847386123, 9.250686732923454744447362339835, 9.636846512157124384757463331095