L(s) = 1 | − i·2-s + i·3-s + 4-s + (−1 − 2i)5-s + 6-s − 4i·7-s − 3i·8-s − 9-s + (−2 + i)10-s + i·12-s + 2i·13-s − 4·14-s + (2 − i)15-s − 16-s − 2i·17-s + i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s + 0.5·4-s + (−0.447 − 0.894i)5-s + 0.408·6-s − 1.51i·7-s − 1.06i·8-s − 0.333·9-s + (−0.632 + 0.316i)10-s + 0.288i·12-s + 0.554i·13-s − 1.06·14-s + (0.516 − 0.258i)15-s − 0.250·16-s − 0.485i·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.646646954\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.646646954\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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
−9.161891937239459487648409481797, −8.200405801661613531070445247007, −7.25108127291580404937172869380, −6.83285221950036323732294950402, −5.40582086246320653379947184613, −4.58133251846812205813242500453, −3.79670714277806958034241819906, −3.14672495861360729195195729473, −1.58297666033820316851832774872, −0.59942514890474629788761004976,
1.76465399518399572789701647427, 2.76117907410350819131260758243, 3.40441335206875611386580634459, 5.27097335197109517589856739534, 5.68801215633961191125092706376, 6.47491528293466444118411478083, 7.28263115555089593740103438497, 7.80270855497288390574074728333, 8.514381935759711504909272011063, 9.411667038227991760366920115256