| L(s) = 1 | + 2.61·5-s + (−1.25 − 2.32i)7-s − 4.29i·11-s − 1.53i·13-s + 0.448·17-s + 1.92i·19-s + 0.737i·23-s + 1.82·25-s − 1.41i·29-s − 6.58i·31-s + (−3.29 − 6.08i)35-s + 2.82·37-s + 6.94·41-s − 9.64·43-s + 2.72·47-s + ⋯ |
| L(s) = 1 | + 1.16·5-s + (−0.475 − 0.879i)7-s − 1.29i·11-s − 0.424i·13-s + 0.108·17-s + 0.442i·19-s + 0.153i·23-s + 0.365·25-s − 0.262i·29-s − 1.18i·31-s + (−0.556 − 1.02i)35-s + 0.464·37-s + 1.08·41-s − 1.47·43-s + 0.397·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.767416315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.767416315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.25 + 2.32i)T \) |
| good | 5 | \( 1 - 2.61T + 5T^{2} \) |
| 11 | \( 1 + 4.29iT - 11T^{2} \) |
| 13 | \( 1 + 1.53iT - 13T^{2} \) |
| 17 | \( 1 - 0.448T + 17T^{2} \) |
| 19 | \( 1 - 1.92iT - 19T^{2} \) |
| 23 | \( 1 - 0.737iT - 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 6.58iT - 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 + 9.64T + 43T^{2} \) |
| 47 | \( 1 - 2.72T + 47T^{2} \) |
| 53 | \( 1 + 2.58iT - 53T^{2} \) |
| 59 | \( 1 + 9.30T + 59T^{2} \) |
| 61 | \( 1 - 8.28iT - 61T^{2} \) |
| 67 | \( 1 - 1.47T + 67T^{2} \) |
| 71 | \( 1 + 11.4iT - 71T^{2} \) |
| 73 | \( 1 + 11.0iT - 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 6.75iT - 97T^{2} \) |
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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
−8.127433182071587601041809183975, −7.55956677550106347516728451413, −6.51413102088172057860593141724, −6.00769500397961318955674445182, −5.47325536228026457595557635322, −4.35185263387810005081784822276, −3.48087017943174635285452809289, −2.71376716068360621146819910049, −1.55439191129801752196315683164, −0.47969363205777208393306780272,
1.49729244474742230443658134247, 2.26443051397656908100982998661, 2.98849411872399286185090565914, 4.23904124583290924981722688537, 5.09739743040148715970824179268, 5.66986632082293728477490958804, 6.56831918390134741881967191158, 6.93474431828726317313584299111, 8.009426615378342004006433850706, 8.899115964433851897970607791879
