L(s) = 1 | + i·2-s − i·3-s − 4-s + (1 − 2i)5-s + 6-s − 5i·7-s − i·8-s − 9-s + (2 + i)10-s − 11-s + i·12-s + 5·14-s + (−2 − i)15-s + 16-s + 4i·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 − 1.88i·7-s − 0.353i·8-s − 0.333·9-s + (0.632 + 0.316i)10-s − 0.301·11-s + 0.288i·12-s + 1.33·14-s + (−0.516 − 0.258i)15-s + 0.250·16-s + 0.970i·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.603105 - 0.975845i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.603105 - 0.975845i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + 5iT - 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 + 7iT - 73T^{2} \) |
| 79 | \( 1 + 15T + 79T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.811493788425270931391302038918, −8.671093947431050969266309820508, −8.040791268105960274438525677327, −7.24458161526450368608660492853, −6.49069888343439444986924777984, −5.55733500803930196949177533093, −4.50740365625074095957041183920, −3.73543790169156736936307645155, −1.76117529757177652857111550953, −0.52648655962142875337134174249,
2.21241908536177274458602143941, 2.69758095415948915636767956929, 3.82969445389007630682392656451, 5.27927439793987147849687201147, 5.65985739249986372350084822645, 6.81544539371626149443485786711, 8.106490942136047456984712173931, 9.027610487362500617128024115490, 9.533836145126046391854071597588, 10.28333496243343160019385547601