| L(s) = 1 | − i·2-s + 4-s + 3i·7-s − 3i·8-s − 11-s + 2i·13-s + 3·14-s − 16-s − 3i·17-s + 19-s + i·22-s + i·23-s + 2·26-s + 3i·28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.5·4-s + 1.13i·7-s − 1.06i·8-s − 0.301·11-s + 0.554i·13-s + 0.801·14-s − 0.250·16-s − 0.727i·17-s + 0.229·19-s + 0.213i·22-s + 0.208i·23-s + 0.392·26-s + 0.566i·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.202883293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.202883293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 11T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 5T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 17iT - 97T^{2} \) |
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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
−9.017041722458801716994923028904, −8.220659555474796358667056744293, −7.28023622010360050458234235152, −6.55760243283327021991045613731, −5.77506754366716819444239097594, −4.89858419998148600373127449043, −3.84998293964607259201682175877, −2.73066940417275293644769737710, −2.34941860132668614952611307400, −1.04255413523686930824297306454,
0.893115655868234040583118614749, 2.24582127773753597897398860312, 3.28092545924390481765675384592, 4.29102506892960570404081863746, 5.20833064497245747081884276596, 6.02171445368533873754317562258, 6.79806317192970297973470887521, 7.34048424876465677186142570469, 8.148868382659839341113182116194, 8.568891059817690710786278515823
