| L(s) = 1 | − i·2-s − 4-s + 2i·5-s + 4i·7-s + i·8-s + 2·10-s + 4i·11-s + 4·14-s + 16-s + 2·17-s + 8i·19-s − 2i·20-s + 4·22-s + 25-s − 4i·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.894i·5-s + 1.51i·7-s + 0.353i·8-s + 0.632·10-s + 1.20i·11-s + 1.06·14-s + 0.250·16-s + 0.485·17-s + 1.83i·19-s − 0.447i·20-s + 0.852·22-s + 0.200·25-s − 0.755i·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.313836230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.313836230\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 16iT - 67T^{2} \) |
| 71 | \( 1 + 8iT - 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 14iT - 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.042753150987512427971315816553, −8.399547444869824165629702392029, −7.49367118316549753137913603084, −6.75882412606412976347109517664, −5.67352763690676226247897342692, −5.31666816703726764753634668059, −4.03167131379461858421417338369, −3.28568409364300103593474278941, −2.31220777880638730092996628525, −1.75061731713770538325996621162,
0.45467745625391820476027404625, 1.16746380394535865093387629751, 2.98752763847768671175599656066, 3.93248086319688664586424341691, 4.65809966085322164903435216152, 5.33017944940951941812404765582, 6.27999142065331352449559284997, 6.98489229009012974886029995967, 7.75408384712726716288203695171, 8.299197635418708122260273390729
