| L(s) = 1 | − 2-s + 4-s + 2.76i·5-s − 7-s − 8-s − 2.76i·10-s + 4.92i·11-s + 2.93i·13-s + 14-s + 16-s + 4.92i·17-s + (1.87 + 3.93i)19-s + 2.76i·20-s − 4.92i·22-s − 0.685i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.23i·5-s − 0.377·7-s − 0.353·8-s − 0.874i·10-s + 1.48i·11-s + 0.815i·13-s + 0.267·14-s + 0.250·16-s + 1.19i·17-s + (0.430 + 0.902i)19-s + 0.618i·20-s − 1.05i·22-s − 0.142i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8967946687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8967946687\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.87 - 3.93i)T \) |
| good | 5 | \( 1 - 2.76iT - 5T^{2} \) |
| 11 | \( 1 - 4.92iT - 11T^{2} \) |
| 13 | \( 1 - 2.93iT - 13T^{2} \) |
| 17 | \( 1 - 4.92iT - 17T^{2} \) |
| 23 | \( 1 + 0.685iT - 23T^{2} \) |
| 29 | \( 1 - 1.65T + 29T^{2} \) |
| 31 | \( 1 + 2.16iT - 31T^{2} \) |
| 37 | \( 1 + 9.77iT - 37T^{2} \) |
| 41 | \( 1 - 2.37T + 41T^{2} \) |
| 43 | \( 1 + 8.15T + 43T^{2} \) |
| 47 | \( 1 - 5.87iT - 47T^{2} \) |
| 53 | \( 1 - 3.47T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 4.26T + 61T^{2} \) |
| 67 | \( 1 - 6.56iT - 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 2.85T + 73T^{2} \) |
| 79 | \( 1 + 11.0iT - 79T^{2} \) |
| 83 | \( 1 - 7.07iT - 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 3.11iT - 97T^{2} \) |
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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
−9.589507755589185074411700083513, −8.558317202227657559988742330957, −7.66648554032458398892347484512, −7.07129849785039560830472319487, −6.50197641194301512057399970669, −5.72452584276918507988063597475, −4.35205452322050406533878839607, −3.54248318247761058001736062612, −2.43889005373008141825572195038, −1.67250080550646288758600010942,
0.43173262069650757805699897322, 1.12575221377167456691436763035, 2.74200825224158117946595404336, 3.44722028829542747549029361082, 4.88360351899177413825221251014, 5.37322446627425756852745324237, 6.34300223773880358276545956783, 7.16814783415401064868657588481, 8.203962363300017330207425952143, 8.532604553892064498836789363972
