| L(s) = 1 | + 2-s + 3-s + 4-s + 2.83·5-s + 6-s + 7-s + 8-s + 9-s + 2.83·10-s + 0.198·11-s + 12-s + 14-s + 2.83·15-s + 16-s + 0.445·17-s + 18-s − 1.01·19-s + 2.83·20-s + 21-s + 0.198·22-s − 5.03·23-s + 24-s + 3.03·25-s + 27-s + 28-s + 4.34·29-s + 2.83·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.26·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.896·10-s + 0.0597·11-s + 0.288·12-s + 0.267·14-s + 0.731·15-s + 0.250·16-s + 0.107·17-s + 0.235·18-s − 0.233·19-s + 0.633·20-s + 0.218·21-s + 0.0422·22-s − 1.05·23-s + 0.204·24-s + 0.606·25-s + 0.192·27-s + 0.188·28-s + 0.806·29-s + 0.517·30-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
| \(L(1)\) |
\(\approx\) |
\(5.812406498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.812406498\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.83T + 5T^{2} \) |
| 11 | \( 1 - 0.198T + 11T^{2} \) |
| 17 | \( 1 - 0.445T + 17T^{2} \) |
| 19 | \( 1 + 1.01T + 19T^{2} \) |
| 23 | \( 1 + 5.03T + 23T^{2} \) |
| 29 | \( 1 - 4.34T + 29T^{2} \) |
| 31 | \( 1 - 2.21T + 31T^{2} \) |
| 37 | \( 1 + 7.43T + 37T^{2} \) |
| 41 | \( 1 + 0.443T + 41T^{2} \) |
| 43 | \( 1 - 7.64T + 43T^{2} \) |
| 47 | \( 1 - 3.63T + 47T^{2} \) |
| 53 | \( 1 - 6.50T + 53T^{2} \) |
| 59 | \( 1 - 5.45T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 5.80T + 71T^{2} \) |
| 73 | \( 1 + 0.641T + 73T^{2} \) |
| 79 | \( 1 - 1.67T + 79T^{2} \) |
| 83 | \( 1 - 6.42T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 9.69T + 97T^{2} \) |
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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
−8.017030762737069606137652934804, −7.01889652385426573935312669833, −6.52556257867114977629399441561, −5.66789716928235038946449469408, −5.25017497592586264527956242862, −4.28207833268321548217227294570, −3.64811352675982167306828202880, −2.50573613104622215107525494693, −2.15495236206106371409399263189, −1.14781709830070000110913214021,
1.14781709830070000110913214021, 2.15495236206106371409399263189, 2.50573613104622215107525494693, 3.64811352675982167306828202880, 4.28207833268321548217227294570, 5.25017497592586264527956242862, 5.66789716928235038946449469408, 6.52556257867114977629399441561, 7.01889652385426573935312669833, 8.017030762737069606137652934804
