| L(s) = 1 | − 0.632i·2-s − 3-s + 1.59·4-s + 0.632i·6-s − 1.76·7-s − 2.27i·8-s + 9-s + 1.39i·11-s − 1.59·12-s + 5.88i·13-s + 1.11i·14-s + 1.75·16-s + (−2.40 + 3.34i)17-s − 0.632i·18-s − 3.54·19-s + ⋯ |
| L(s) = 1 | − 0.447i·2-s − 0.577·3-s + 0.799·4-s + 0.258i·6-s − 0.666·7-s − 0.805i·8-s + 0.333·9-s + 0.420i·11-s − 0.461·12-s + 1.63i·13-s + 0.297i·14-s + 0.439·16-s + (−0.583 + 0.812i)17-s − 0.149i·18-s − 0.814·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9358178118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9358178118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + (2.40 - 3.34i)T \) |
| good | 2 | \( 1 + 0.632iT - 2T^{2} \) |
| 7 | \( 1 + 1.76T + 7T^{2} \) |
| 11 | \( 1 - 1.39iT - 11T^{2} \) |
| 13 | \( 1 - 5.88iT - 13T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 23 | \( 1 + 8.78T + 23T^{2} \) |
| 29 | \( 1 - 3.51iT - 29T^{2} \) |
| 31 | \( 1 + 0.844iT - 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 - 7.55iT - 41T^{2} \) |
| 43 | \( 1 - 5.55iT - 43T^{2} \) |
| 47 | \( 1 + 7.83iT - 47T^{2} \) |
| 53 | \( 1 - 2.93iT - 53T^{2} \) |
| 59 | \( 1 - 1.95T + 59T^{2} \) |
| 61 | \( 1 - 7.10iT - 61T^{2} \) |
| 67 | \( 1 - 1.45iT - 67T^{2} \) |
| 71 | \( 1 + 9.90iT - 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 11.6iT - 79T^{2} \) |
| 83 | \( 1 - 3.25iT - 83T^{2} \) |
| 89 | \( 1 + 0.725T + 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.966775685460247418541190392106, −9.366205672223540531700662763727, −8.195590606347529101483605842059, −7.17791771196608262968819931522, −6.32675115179195547343466653830, −6.15025844646258363665347952286, −4.47454717996846374423031046400, −3.87334001565990337587636384814, −2.44674274967718857138018370571, −1.59573118079810131457125003843,
0.38781368216802929212024937338, 2.22453892495906374537158336270, 3.20271287029610671519875821128, 4.46268675267086554450017496376, 5.80891072970258042667896819538, 5.94803217782505373354909731012, 6.92894572393919323470559672421, 7.77227133016116232177181149333, 8.414501725905742971784280130848, 9.676422124812002341040288977408
