| L(s) = 1 | + (2 + 2i)5-s + i·7-s + 3i·9-s + (−1 − i)11-s − 2·17-s + (−2 + 2i)19-s + 6i·23-s + 3i·25-s + (7 − 7i)29-s + 8·31-s + (−2 + 2i)35-s + (−5 − 5i)37-s + 10i·41-s + (1 + i)43-s + (−6 + 6i)45-s + ⋯ |
| L(s) = 1 | + (0.894 + 0.894i)5-s + 0.377i·7-s + i·9-s + (−0.301 − 0.301i)11-s − 0.485·17-s + (−0.458 + 0.458i)19-s + 1.25i·23-s + 0.600i·25-s + (1.29 − 1.29i)29-s + 1.43·31-s + (−0.338 + 0.338i)35-s + (−0.821 − 0.821i)37-s + 1.56i·41-s + (0.152 + 0.152i)43-s + (−0.894 + 0.894i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26564 + 0.845677i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26564 + 0.845677i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - 3iT^{2} \) |
| 5 | \( 1 + (-2 - 2i)T + 5iT^{2} \) |
| 11 | \( 1 + (1 + i)T + 11iT^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + (2 - 2i)T - 19iT^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + (-7 + 7i)T - 29iT^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (5 + 5i)T + 37iT^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + (-1 - i)T + 43iT^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + (1 + i)T + 53iT^{2} \) |
| 59 | \( 1 + (8 + 8i)T + 59iT^{2} \) |
| 61 | \( 1 + (-6 + 6i)T - 61iT^{2} \) |
| 67 | \( 1 + (3 - 3i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + (-10 + 10i)T - 83iT^{2} \) |
| 89 | \( 1 + 14iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−11.09175911822384893501823113793, −10.35670127096813347327475871805, −9.702580643477614558712439334401, −8.488434702398828304283790782527, −7.63535624886946463675416202902, −6.43597228751578873118889251037, −5.76115276136681354195826822349, −4.58390955543127375685342795580, −2.93871229315693150871133454447, −2.03452628303811696987370072213,
1.01412491854280694197179319487, 2.60887290518471560943107486435, 4.22433895433586790162957033536, 5.11215983379857543480039693700, 6.28252593000336441674969014597, 7.03215059120741030488718027095, 8.589862870716613432622908501128, 8.965539802145006168766497355833, 10.07397152515153498375012293207, 10.67805717949875776131921745508
