| L(s) = 1 | + 3i·3-s + 2i·7-s − 6·9-s − 11-s + 4i·13-s − 5i·17-s + 19-s − 6·21-s + 2i·23-s − 9i·27-s + 8·29-s − 10·31-s − 3i·33-s + 6i·37-s − 12·39-s + ⋯ |
| L(s) = 1 | + 1.73i·3-s + 0.755i·7-s − 2·9-s − 0.301·11-s + 1.10i·13-s − 1.21i·17-s + 0.229·19-s − 1.30·21-s + 0.417i·23-s − 1.73i·27-s + 1.48·29-s − 1.79·31-s − 0.522i·33-s + 0.986i·37-s − 1.92·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.262733 + 1.11295i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.262733 + 1.11295i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 5iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 3iT - 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 13iT - 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−11.49872412557514933520958907730, −10.69825059377453912838744929628, −9.669487069599298121277040433615, −9.229689823562002455895683396539, −8.361311322866435477509894911609, −6.87854938788815571911988126649, −5.50901059185254097494787571248, −4.86836874346180885770841587027, −3.77178671360293152835645872175, −2.59587241429448041749976204684,
0.75402159991910623317391656143, 2.16011893007561993304510332556, 3.56636821323802014617499659948, 5.33768917294660560120686567234, 6.33003125406223115386590034977, 7.19887754643194255047378783871, 7.939961448700414459634292619666, 8.635798442302891806468299240118, 10.21040167473816967659040112772, 10.94222156935275510192231394945
