| L(s) = 1 | − 2.37·3-s + 7-s + 2.62·9-s + 6.37·11-s − 4.37·13-s + 0.372·17-s − 4.74·19-s − 2.37·21-s + 4.74·23-s + 0.883·27-s − 4.37·29-s − 8·31-s − 15.1·33-s + 2·37-s + 10.3·39-s + 6.74·41-s + 8.74·43-s + 7.11·47-s + 49-s − 0.883·51-s − 10.7·53-s + 11.2·57-s + 8·59-s − 2.74·61-s + 2.62·63-s + 4·67-s − 11.2·69-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 0.377·7-s + 0.875·9-s + 1.92·11-s − 1.21·13-s + 0.0902·17-s − 1.08·19-s − 0.517·21-s + 0.989·23-s + 0.169·27-s − 0.811·29-s − 1.43·31-s − 2.63·33-s + 0.328·37-s + 1.66·39-s + 1.05·41-s + 1.33·43-s + 1.03·47-s + 0.142·49-s − 0.123·51-s − 1.47·53-s + 1.49·57-s + 1.04·59-s − 0.351·61-s + 0.331·63-s + 0.488·67-s − 1.35·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.036787562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.036787562\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 11 | \( 1 - 6.37T + 11T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 - 4.74T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6.74T + 41T^{2} \) |
| 43 | \( 1 - 8.74T + 43T^{2} \) |
| 47 | \( 1 - 7.11T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 2.74T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 9.86T + 97T^{2} \) |
| 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
−9.397991707027163850618833447413, −9.109179329456040175650741661490, −7.72950235521558262835445175074, −6.91045554305168647134119697223, −6.28683622552320494088663864291, −5.43099998452876702819854300790, −4.61292101077022212155390767498, −3.79847429204527997960070107939, −2.11361203192552443151639818170, −0.800014161814407391550046830876,
0.800014161814407391550046830876, 2.11361203192552443151639818170, 3.79847429204527997960070107939, 4.61292101077022212155390767498, 5.43099998452876702819854300790, 6.28683622552320494088663864291, 6.91045554305168647134119697223, 7.72950235521558262835445175074, 9.109179329456040175650741661490, 9.397991707027163850618833447413
