L(s) = 1 | + 3i·5-s − 3·7-s − 3i·11-s − 6i·13-s + 6·17-s + 2i·19-s − 6·23-s − 4·25-s − 6i·29-s − 3·31-s − 9i·35-s + 6i·37-s + 6·41-s − 8i·43-s + 12·47-s + ⋯ |
L(s) = 1 | + 1.34i·5-s − 1.13·7-s − 0.904i·11-s − 1.66i·13-s + 1.45·17-s + 0.458i·19-s − 1.25·23-s − 0.800·25-s − 1.11i·29-s − 0.538·31-s − 1.52i·35-s + 0.986i·37-s + 0.937·41-s − 1.21i·43-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.212995121\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.212995121\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3iT - 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 3iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
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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
−9.480043945550055346963208552775, −8.145833724647952143060398246561, −7.74589077067989058948409695891, −6.74334493425676065742555534630, −5.93882696173945048865786024819, −5.57415611565190300563102268594, −3.70250198959402080384251256898, −3.32383245322115079032326378990, −2.48525380783069184067816505460, −0.52179338862785849113736394077,
1.12795674379503004474024297455, 2.28989152185708834096128242050, 3.72430536914905413762294887999, 4.39632025619609821728753645585, 5.31906052874823733325065728115, 6.16737532495647532397136123949, 7.07511010345667259692045125412, 7.80322614008789509086620237296, 8.930917743310760361691679974891, 9.361437141796803309891765160369