| L(s) = 1 | − 2-s + 1.19·3-s + 4-s − 2.46·5-s − 1.19·6-s − 0.891·7-s − 8-s − 1.56·9-s + 2.46·10-s − 4.56·11-s + 1.19·12-s + 2.98·13-s + 0.891·14-s − 2.95·15-s + 16-s − 7.59·17-s + 1.56·18-s − 0.542·19-s − 2.46·20-s − 1.06·21-s + 4.56·22-s − 0.187·23-s − 1.19·24-s + 1.05·25-s − 2.98·26-s − 5.47·27-s − 0.891·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.692·3-s + 0.5·4-s − 1.10·5-s − 0.489·6-s − 0.336·7-s − 0.353·8-s − 0.520·9-s + 0.778·10-s − 1.37·11-s + 0.346·12-s + 0.827·13-s + 0.238·14-s − 0.762·15-s + 0.250·16-s − 1.84·17-s + 0.368·18-s − 0.124·19-s − 0.550·20-s − 0.233·21-s + 0.973·22-s − 0.0390·23-s − 0.244·24-s + 0.211·25-s − 0.584·26-s − 1.05·27-s − 0.168·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4727597245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4727597245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 + 2.46T + 5T^{2} \) |
| 7 | \( 1 + 0.891T + 7T^{2} \) |
| 11 | \( 1 + 4.56T + 11T^{2} \) |
| 13 | \( 1 - 2.98T + 13T^{2} \) |
| 17 | \( 1 + 7.59T + 17T^{2} \) |
| 19 | \( 1 + 0.542T + 19T^{2} \) |
| 23 | \( 1 + 0.187T + 23T^{2} \) |
| 29 | \( 1 - 1.54T + 29T^{2} \) |
| 37 | \( 1 - 8.76T + 37T^{2} \) |
| 41 | \( 1 - 1.00T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 8.62T + 53T^{2} \) |
| 59 | \( 1 + 4.95T + 59T^{2} \) |
| 61 | \( 1 - 5.35T + 61T^{2} \) |
| 67 | \( 1 - 6.65T + 67T^{2} \) |
| 71 | \( 1 + 6.28T + 71T^{2} \) |
| 73 | \( 1 - 3.07T + 73T^{2} \) |
| 79 | \( 1 - 9.36T + 79T^{2} \) |
| 83 | \( 1 + 0.634T + 83T^{2} \) |
| 89 | \( 1 + 7.59T + 89T^{2} \) |
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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
−8.148499371014999916522600595823, −7.73999476017845293685404134797, −6.80069116200971637083030507338, −6.21312415701294092666881838908, −5.18068879295774494433361199689, −4.26622803973354451596967529707, −3.43557029522707722059632299692, −2.76320183181581147044959010527, −1.97946897919584595319161616622, −0.36118340919128258882988763604,
0.36118340919128258882988763604, 1.97946897919584595319161616622, 2.76320183181581147044959010527, 3.43557029522707722059632299692, 4.26622803973354451596967529707, 5.18068879295774494433361199689, 6.21312415701294092666881838908, 6.80069116200971637083030507338, 7.73999476017845293685404134797, 8.148499371014999916522600595823
