| L(s) = 1 | + 2-s − 3-s + 4-s − 2.41·5-s − 6-s + 7-s + 8-s + 9-s − 2.41·10-s + 2.07·11-s − 12-s + 14-s + 2.41·15-s + 16-s + 6.31·17-s + 18-s − 1.82·19-s − 2.41·20-s − 21-s + 2.07·22-s − 2.11·23-s − 24-s + 0.833·25-s − 27-s + 28-s − 4.51·29-s + 2.41·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.08·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.763·10-s + 0.624·11-s − 0.288·12-s + 0.267·14-s + 0.623·15-s + 0.250·16-s + 1.53·17-s + 0.235·18-s − 0.418·19-s − 0.540·20-s − 0.218·21-s + 0.441·22-s − 0.440·23-s − 0.204·24-s + 0.166·25-s − 0.192·27-s + 0.188·28-s − 0.839·29-s + 0.440·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.304009121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.304009121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2.41T + 5T^{2} \) |
| 11 | \( 1 - 2.07T + 11T^{2} \) |
| 17 | \( 1 - 6.31T + 17T^{2} \) |
| 19 | \( 1 + 1.82T + 19T^{2} \) |
| 23 | \( 1 + 2.11T + 23T^{2} \) |
| 29 | \( 1 + 4.51T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 6.82T + 41T^{2} \) |
| 43 | \( 1 + 1.29T + 43T^{2} \) |
| 47 | \( 1 - 3.78T + 47T^{2} \) |
| 53 | \( 1 + 0.389T + 53T^{2} \) |
| 59 | \( 1 + 6.22T + 59T^{2} \) |
| 61 | \( 1 - 5.20T + 61T^{2} \) |
| 67 | \( 1 - 9.51T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 8.31T + 73T^{2} \) |
| 79 | \( 1 - 3.45T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 5.26T + 89T^{2} \) |
| 97 | \( 1 - 9.97T + 97T^{2} \) |
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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
−7.963583040631914173577201170894, −7.10674450932167057147473899091, −6.54019726627635344659542082230, −5.67366273562666828326691832799, −5.14624830247316024393632313430, −4.18590173707445489848873349130, −3.87298438653553272499080702237, −2.97360505260528910309476128299, −1.76020206824708740214208942485, −0.73368771961242468556073543236,
0.73368771961242468556073543236, 1.76020206824708740214208942485, 2.97360505260528910309476128299, 3.87298438653553272499080702237, 4.18590173707445489848873349130, 5.14624830247316024393632313430, 5.67366273562666828326691832799, 6.54019726627635344659542082230, 7.10674450932167057147473899091, 7.963583040631914173577201170894
