| L(s) = 1 | − 2.29i·3-s + 1.61i·7-s − 2.26·9-s − 2.79·11-s + 4.88i·13-s − 3.54i·17-s + 2.79·19-s + 3.71·21-s − i·23-s − 1.67i·27-s + 3.36·29-s − 4.84·31-s + 6.41i·33-s − 3.87i·37-s + 11.2·39-s + ⋯ |
| L(s) = 1 | − 1.32i·3-s + 0.611i·7-s − 0.756·9-s − 0.841·11-s + 1.35i·13-s − 0.859i·17-s + 0.640·19-s + 0.810·21-s − 0.208i·23-s − 0.322i·27-s + 0.624·29-s − 0.870·31-s + 1.11i·33-s − 0.636i·37-s + 1.79·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.125337225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.125337225\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + 2.29iT - 3T^{2} \) |
| 7 | \( 1 - 1.61iT - 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 - 4.88iT - 13T^{2} \) |
| 17 | \( 1 + 3.54iT - 17T^{2} \) |
| 19 | \( 1 - 2.79T + 19T^{2} \) |
| 29 | \( 1 - 3.36T + 29T^{2} \) |
| 31 | \( 1 + 4.84T + 31T^{2} \) |
| 37 | \( 1 + 3.87iT - 37T^{2} \) |
| 41 | \( 1 + 0.327T + 41T^{2} \) |
| 43 | \( 1 + 5.38iT - 43T^{2} \) |
| 47 | \( 1 + 0.0121iT - 47T^{2} \) |
| 53 | \( 1 + 0.866iT - 53T^{2} \) |
| 59 | \( 1 + 2.96T + 59T^{2} \) |
| 61 | \( 1 + 7.29T + 61T^{2} \) |
| 67 | \( 1 + 6.94iT - 67T^{2} \) |
| 71 | \( 1 + 2.16T + 71T^{2} \) |
| 73 | \( 1 + 6.02iT - 73T^{2} \) |
| 79 | \( 1 + 6.50T + 79T^{2} \) |
| 83 | \( 1 + 7.88iT - 83T^{2} \) |
| 89 | \( 1 - 9.71T + 89T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
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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
−7.75446773825665972165113640335, −7.33656284199830328089725449773, −6.69316731493012397603906848194, −5.94629023593061486593002913493, −5.20581101976438886077993914329, −4.35043054540193119789288563592, −3.10251640569607887537206814164, −2.29668831547900638599572479695, −1.62136978557804193686578423208, −0.32639880268806939173996251056,
1.14823470356046305133745746349, 2.69208264437826901328369703911, 3.42389886805738782261799699717, 4.08559720036746945237901444015, 4.97204842797564383403903496117, 5.43438037979987586785040853670, 6.28018374971027179379741381585, 7.41063177313916756808673474522, 7.904182214488956606676597863167, 8.676691279004127631465036000806
