| L(s) = 1 | + (−1.41 + 1.41i)5-s − 7-s + (2.82 + 2.82i)11-s + (4 − 4i)13-s − 5.65i·17-s + (−4 − 4i)19-s + 1.41i·23-s + 0.999i·25-s + (5.65 + 5.65i)29-s + (1.41 − 1.41i)35-s + (−5 − 5i)37-s − 8.48·41-s + (−7 + 7i)43-s + 49-s + (7.07 − 7.07i)53-s + ⋯ |
| L(s) = 1 | + (−0.632 + 0.632i)5-s − 0.377·7-s + (0.852 + 0.852i)11-s + (1.10 − 1.10i)13-s − 1.37i·17-s + (−0.917 − 0.917i)19-s + 0.294i·23-s + 0.199i·25-s + (1.05 + 1.05i)29-s + (0.239 − 0.239i)35-s + (−0.821 − 0.821i)37-s − 1.32·41-s + (−1.06 + 1.06i)43-s + 0.142·49-s + (0.971 − 0.971i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.314611907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.314611907\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 5 | \( 1 + (1.41 - 1.41i)T - 5iT^{2} \) |
| 11 | \( 1 + (-2.82 - 2.82i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4 + 4i)T - 13iT^{2} \) |
| 17 | \( 1 + 5.65iT - 17T^{2} \) |
| 19 | \( 1 + (4 + 4i)T + 19iT^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 + (-5.65 - 5.65i)T + 29iT^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (5 + 5i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 + (7 - 7i)T - 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (-7.07 + 7.07i)T - 53iT^{2} \) |
| 59 | \( 1 + (5.65 + 5.65i)T + 59iT^{2} \) |
| 61 | \( 1 + (-4 + 4i)T - 61iT^{2} \) |
| 67 | \( 1 + (7 + 7i)T + 67iT^{2} \) |
| 71 | \( 1 + 7.07iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + (2.82 - 2.82i)T - 83iT^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
| 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
−8.365253145427205286465183319587, −7.44504385223513657041967225781, −6.83920849739247339488249999788, −6.40687392701975775371172113411, −5.23321677984293696802096041929, −4.55911885750188444370028787459, −3.41714937847170717130933719313, −3.14676676469055961882837740629, −1.80215997049526176719355365312, −0.44751470645164086790801955248,
1.02099167244626558750972894805, 1.93683646412438082399957755095, 3.42574705865315681205565319167, 3.98686072008818576098137874708, 4.50396442657349809579027122989, 5.86311682642044434272296251452, 6.28726590392987879590907324833, 6.92467017427390391744015989106, 8.200594153576711322585734028385, 8.619451005254685631258533101422
