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 | \( 1 - 8iT - 97T^{2} \) |
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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
−9.411667038227991760366920115256, −8.514381935759711504909272011063, −7.80270855497288390574074728333, −7.28263115555089593740103438497, −6.47491528293466444118411478083, −5.68801215633961191125092706376, −5.27097335197109517589856739534, −3.40441335206875611386580634459, −2.76117907410350819131260758243, −1.76465399518399572789701647427,
0.59942514890474629788761004976, 1.58297666033820316851832774872, 3.14672495861360729195195729473, 3.79670714277806958034241819906, 4.58133251846812205813242500453, 5.40582086246320653379947184613, 6.83285221950036323732294950402, 7.25108127291580404937172869380, 8.200405801661613531070445247007, 9.161891937239459487648409481797