| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 0.643·7-s − 8-s + 9-s − 10-s + 4.40·11-s + 12-s − 0.643·14-s + 15-s + 16-s + 0.939·17-s − 18-s + 2.75·19-s + 20-s + 0.643·21-s − 4.40·22-s − 2.04·23-s − 24-s + 25-s + 27-s + 0.643·28-s − 0.149·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.243·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.32·11-s + 0.288·12-s − 0.171·14-s + 0.258·15-s + 0.250·16-s + 0.227·17-s − 0.235·18-s + 0.631·19-s + 0.223·20-s + 0.140·21-s − 0.939·22-s − 0.427·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.121·28-s − 0.0276·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.349877264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.349877264\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 0.643T + 7T^{2} \) |
| 11 | \( 1 - 4.40T + 11T^{2} \) |
| 17 | \( 1 - 0.939T + 17T^{2} \) |
| 19 | \( 1 - 2.75T + 19T^{2} \) |
| 23 | \( 1 + 2.04T + 23T^{2} \) |
| 29 | \( 1 + 0.149T + 29T^{2} \) |
| 31 | \( 1 - 2.08T + 31T^{2} \) |
| 37 | \( 1 - 6.07T + 37T^{2} \) |
| 41 | \( 1 + 5.74T + 41T^{2} \) |
| 43 | \( 1 - 4.69T + 43T^{2} \) |
| 47 | \( 1 - 7T + 47T^{2} \) |
| 53 | \( 1 - 1.68T + 53T^{2} \) |
| 59 | \( 1 - 7.85T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 1.45T + 67T^{2} \) |
| 71 | \( 1 + 7.03T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 0.929T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 7.47T + 89T^{2} \) |
| 97 | \( 1 + 5.05T + 97T^{2} \) |
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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
−8.267255181571834956686076032174, −7.67588178284278394494966457092, −6.87655426743959361921462659281, −6.27820479640616887958545950680, −5.45509955498287822412439688347, −4.40669639726121348856024649439, −3.60553907787654972232495417178, −2.69177342298931181800194980305, −1.75967656300857889110141940812, −0.973173877143817496375904571229,
0.973173877143817496375904571229, 1.75967656300857889110141940812, 2.69177342298931181800194980305, 3.60553907787654972232495417178, 4.40669639726121348856024649439, 5.45509955498287822412439688347, 6.27820479640616887958545950680, 6.87655426743959361921462659281, 7.67588178284278394494966457092, 8.267255181571834956686076032174
