| L(s) = 1 | − i·2-s − 4-s + 1.81·5-s + (−0.812 − 2.51i)7-s + i·8-s − 1.81i·10-s + 6.05i·11-s + 2.51i·13-s + (−2.51 + 0.812i)14-s + 16-s + 4.50·17-s − 4.56i·19-s − 1.81·20-s + 6.05·22-s + 6.67i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.813·5-s + (−0.307 − 0.951i)7-s + 0.353i·8-s − 0.575i·10-s + 1.82i·11-s + 0.696i·13-s + (−0.672 + 0.217i)14-s + 0.250·16-s + 1.09·17-s − 1.04i·19-s − 0.406·20-s + 1.29·22-s + 1.39i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.329210699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.329210699\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (0.812 + 2.51i)T \) |
| 29 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.81T + 5T^{2} \) |
| 11 | \( 1 - 6.05iT - 11T^{2} \) |
| 13 | \( 1 - 2.51iT - 13T^{2} \) |
| 17 | \( 1 - 4.50T + 17T^{2} \) |
| 19 | \( 1 + 4.56iT - 19T^{2} \) |
| 23 | \( 1 - 6.67iT - 23T^{2} \) |
| 31 | \( 1 - 7.40iT - 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 + 8.46T + 41T^{2} \) |
| 43 | \( 1 + 5.99T + 43T^{2} \) |
| 47 | \( 1 + 5.40T + 47T^{2} \) |
| 53 | \( 1 + 0.309iT - 53T^{2} \) |
| 59 | \( 1 + 3.41T + 59T^{2} \) |
| 61 | \( 1 - 10.0iT - 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 2.60iT - 71T^{2} \) |
| 73 | \( 1 - 6.17iT - 73T^{2} \) |
| 79 | \( 1 + 9.93T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 9.92iT - 97T^{2} \) |
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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
−8.964865662471032788699063561020, −7.84200911711015567449571625088, −7.06865081758180632562199755333, −6.65252527173149887344556889491, −5.34663673398900973128032123595, −4.85224602243825273604692412047, −3.92058788192555424904240637382, −3.13235632736217866204525418862, −1.92641923792098924653854245295, −1.38690789216272941609062315514,
0.38232193995889270996558418691, 1.81511312725496731808096730507, 3.05373591803734834122239433771, 3.58587621793803546223209564655, 5.03679953204046827453416762109, 5.69951322066116541919974141326, 6.00595028065552017087615276499, 6.68161593466222055255919871405, 7.979122803374727808561148146982, 8.322654480378292179682634809426
