| L(s) = 1 | − i·3-s − i·7-s − 9-s + 11-s + 4i·13-s − 2i·17-s + 4·19-s − 21-s + 7i·23-s + i·27-s + 9·29-s − 2·31-s − i·33-s − i·37-s + 4·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 0.301·11-s + 1.10i·13-s − 0.485i·17-s + 0.917·19-s − 0.218·21-s + 1.45i·23-s + 0.192i·27-s + 1.67·29-s − 0.359·31-s − 0.174i·33-s − 0.164i·37-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.812408588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.812408588\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 7iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 9iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 9iT - 67T^{2} \) |
| 71 | \( 1 - 5T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 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
−9.146465870184735332934552115827, −8.188741486731160862230965141119, −7.32299928303650050548369993068, −6.90661490351405145622428119943, −5.96931559488246018620684285268, −5.08580509834219888984809235677, −4.10716289327802357148768155796, −3.16746886683307598405410565243, −1.99243152912656953148516874951, −0.926719901077701485248323616985,
0.907060684603556304485651327637, 2.52870192577483275079273184112, 3.27550462808842902712028244703, 4.35614797214547920585632639742, 5.10506954473409917021379668141, 5.97022906938966719531904234320, 6.66280959340880125256554989821, 7.86820580502641555615456375265, 8.351351856227891143231312056968, 9.220977692434289125813218020240
