| L(s) = 1 | − i·2-s − 4-s + 0.802·5-s + (−0.522 − 2.59i)7-s + i·8-s − 0.802i·10-s + 4.88i·11-s + 3.72i·13-s + (−2.59 + 0.522i)14-s + 16-s − 2.59·17-s − i·19-s − 0.802·20-s + 4.88·22-s − 6.06i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.358·5-s + (−0.197 − 0.980i)7-s + 0.353i·8-s − 0.253i·10-s + 1.47i·11-s + 1.03i·13-s + (−0.693 + 0.139i)14-s + 0.250·16-s − 0.628·17-s − 0.229i·19-s − 0.179·20-s + 1.04·22-s − 1.26i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4161378755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4161378755\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.522 + 2.59i)T \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 - 0.802T + 5T^{2} \) |
| 11 | \( 1 - 4.88iT - 11T^{2} \) |
| 13 | \( 1 - 3.72iT - 13T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 23 | \( 1 + 6.06iT - 23T^{2} \) |
| 29 | \( 1 + 3.01iT - 29T^{2} \) |
| 31 | \( 1 + 9.34iT - 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 9.75T + 41T^{2} \) |
| 43 | \( 1 + 9.95T + 43T^{2} \) |
| 47 | \( 1 - 3.04T + 47T^{2} \) |
| 53 | \( 1 + 7.65iT - 53T^{2} \) |
| 59 | \( 1 - 9.82T + 59T^{2} \) |
| 61 | \( 1 + 4.40iT - 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 5.62iT - 71T^{2} \) |
| 73 | \( 1 + 3.43iT - 73T^{2} \) |
| 79 | \( 1 + 3.46T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 0.544T + 89T^{2} \) |
| 97 | \( 1 - 5.78iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.703832095112358162651131503189, −7.78038431556363601790777778420, −6.89573329502493663606859845783, −6.41299906133914141613394776900, −5.03749534141334440828433743476, −4.36363397815929110325390938890, −3.76608286325568319331299794584, −2.33684058871514102794898565299, −1.74351932540259616188753095798, −0.13340387958430038035172308672,
1.54829586770773952228835613880, 3.00402819471380469309737063183, 3.57518076934732680919296398765, 5.11342047609522916599560546391, 5.52703996457030846823001524902, 6.18409616114480310358901235687, 6.98737361763186074597637642575, 8.013587676685969242496988535838, 8.632658384388326497212240235406, 9.068830364151651349033634528654
