| L(s) = 1 | + 2-s + 4-s + 2.69i·5-s + 7-s + 8-s + 2.69i·10-s + 4.10i·11-s + 2.15i·13-s + 14-s + 16-s − 6.93i·17-s + (3.42 + 2.69i)19-s + 2.69i·20-s + 4.10i·22-s − 6.39i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.20i·5-s + 0.377·7-s + 0.353·8-s + 0.851i·10-s + 1.23i·11-s + 0.598i·13-s + 0.267·14-s + 0.250·16-s − 1.68i·17-s + (0.786 + 0.617i)19-s + 0.601i·20-s + 0.875i·22-s − 1.33i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0500 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0500 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.892258521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.892258521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + (-3.42 - 2.69i)T \) |
| good | 5 | \( 1 - 2.69iT - 5T^{2} \) |
| 11 | \( 1 - 4.10iT - 11T^{2} \) |
| 13 | \( 1 - 2.15iT - 13T^{2} \) |
| 17 | \( 1 + 6.93iT - 17T^{2} \) |
| 23 | \( 1 + 6.39iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 8.55iT - 31T^{2} \) |
| 37 | \( 1 - 10.5iT - 37T^{2} \) |
| 41 | \( 1 + 7.05T + 41T^{2} \) |
| 43 | \( 1 - 3.05T + 43T^{2} \) |
| 47 | \( 1 - 6.26iT - 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 2.75T + 59T^{2} \) |
| 61 | \( 1 + 5.80T + 61T^{2} \) |
| 67 | \( 1 + 0.136iT - 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 2.56T + 73T^{2} \) |
| 79 | \( 1 + 7.54iT - 79T^{2} \) |
| 83 | \( 1 + 4.84iT - 83T^{2} \) |
| 89 | \( 1 + 3.05T + 89T^{2} \) |
| 97 | \( 1 - 7.19iT - 97T^{2} \) |
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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
−9.287920854037948394573799975762, −8.183645416990568618308270532109, −7.24263747325920116262718621175, −6.90162643323386971657010123210, −6.19579973586348294906105016192, −4.81267622247679463466839598560, −4.68938487651155240154129785821, −3.23307296101242430698602357279, −2.71153747618484836171510788021, −1.57433447194159680114447710149,
0.78520609859902613960977065827, 1.83515687763942055787382403852, 3.20994598069921161391330062738, 3.95611487375684778536171562191, 4.86678054340306352762910919203, 5.66233208384225124359612604958, 6.00384330553721909267159466684, 7.32254845701765733690596330532, 8.100950858243526210440137746472, 8.616123966409622034152518379520
