| L(s) = 1 | − 2-s + 3.05·3-s + 4-s + 1.46·5-s − 3.05·6-s − 8-s + 6.32·9-s − 1.46·10-s − 1.18·11-s + 3.05·12-s + 13-s + 4.46·15-s + 16-s + 6.51·17-s − 6.32·18-s − 4·19-s + 1.46·20-s + 1.18·22-s − 1.73·23-s − 3.05·24-s − 2.86·25-s − 26-s + 10.1·27-s + 1.18·29-s − 4.46·30-s + 6.73·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.76·3-s + 0.5·4-s + 0.653·5-s − 1.24·6-s − 0.353·8-s + 2.10·9-s − 0.461·10-s − 0.357·11-s + 0.881·12-s + 0.277·13-s + 1.15·15-s + 0.250·16-s + 1.58·17-s − 1.49·18-s − 0.917·19-s + 0.326·20-s + 0.253·22-s − 0.361·23-s − 0.623·24-s − 0.573·25-s − 0.196·26-s + 1.95·27-s + 0.220·29-s − 0.814·30-s + 1.20·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.580087874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.580087874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 3.05T + 3T^{2} \) |
| 5 | \( 1 - 1.46T + 5T^{2} \) |
| 11 | \( 1 + 1.18T + 11T^{2} \) |
| 17 | \( 1 - 6.51T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 1.73T + 23T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 + 8.89T + 37T^{2} \) |
| 41 | \( 1 - 2.92T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 5.32T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 + 7.70T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 6.76T + 97T^{2} \) |
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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
−9.547869049417696197720550393339, −8.896296109359886186656550676649, −8.051350744296809222866576432789, −7.72117083876957512042360433045, −6.62363019414199984194242485129, −5.63643676309001835091498660886, −4.21096786098111560195417268153, −3.18429849019966966561897933393, −2.37528495387793478100178885968, −1.41655737864030535304798372541,
1.41655737864030535304798372541, 2.37528495387793478100178885968, 3.18429849019966966561897933393, 4.21096786098111560195417268153, 5.63643676309001835091498660886, 6.62363019414199984194242485129, 7.72117083876957512042360433045, 8.051350744296809222866576432789, 8.896296109359886186656550676649, 9.547869049417696197720550393339
