| L(s) = 1 | + 6.28e18·2-s + 2.54e29·3-s + 2.88e37·4-s − 1.37e43·5-s + 1.59e48·6-s + 5.88e51·7-s + 1.14e56·8-s + 1.60e58·9-s − 8.61e61·10-s + 8.14e63·11-s + 7.32e66·12-s + 4.84e68·13-s + 3.69e70·14-s − 3.48e72·15-s + 4.10e74·16-s + 1.33e75·17-s + 1.00e77·18-s − 1.53e78·19-s − 3.95e80·20-s + 1.49e81·21-s + 5.11e82·22-s − 3.99e83·23-s + 2.90e85·24-s + 9.40e85·25-s + 3.04e87·26-s − 8.25e87·27-s + 1.69e89·28-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 1.15·3-s + 2.70·4-s − 1.41·5-s + 2.22·6-s + 0.625·7-s + 3.29·8-s + 0.330·9-s − 2.72·10-s + 0.733·11-s + 3.12·12-s + 1.50·13-s + 1.20·14-s − 1.63·15-s + 3.63·16-s + 0.283·17-s + 0.637·18-s − 0.348·19-s − 3.83·20-s + 0.721·21-s + 1.41·22-s − 0.716·23-s + 3.79·24-s + 1.00·25-s + 2.90·26-s − 0.772·27-s + 1.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(124-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+61.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(62)\) |
\(\approx\) |
\(11.55978426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.55978426\) |
| \(L(\frac{125}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 6.28e18T + 1.06e37T^{2} \) |
| 3 | \( 1 - 2.54e29T + 4.85e58T^{2} \) |
| 5 | \( 1 + 1.37e43T + 9.40e85T^{2} \) |
| 7 | \( 1 - 5.88e51T + 8.85e103T^{2} \) |
| 11 | \( 1 - 8.14e63T + 1.23e128T^{2} \) |
| 13 | \( 1 - 4.84e68T + 1.03e137T^{2} \) |
| 17 | \( 1 - 1.33e75T + 2.21e151T^{2} \) |
| 19 | \( 1 + 1.53e78T + 1.93e157T^{2} \) |
| 23 | \( 1 + 3.99e83T + 3.10e167T^{2} \) |
| 29 | \( 1 + 8.68e89T + 7.49e179T^{2} \) |
| 31 | \( 1 - 4.77e91T + 2.73e183T^{2} \) |
| 37 | \( 1 - 2.08e96T + 7.74e192T^{2} \) |
| 41 | \( 1 - 1.76e99T + 2.35e198T^{2} \) |
| 43 | \( 1 - 9.95e99T + 8.25e200T^{2} \) |
| 47 | \( 1 - 3.47e102T + 4.65e205T^{2} \) |
| 53 | \( 1 - 1.95e106T + 1.21e212T^{2} \) |
| 59 | \( 1 - 7.27e108T + 6.52e217T^{2} \) |
| 61 | \( 1 + 3.89e109T + 3.94e219T^{2} \) |
| 67 | \( 1 + 4.59e111T + 4.04e224T^{2} \) |
| 71 | \( 1 + 8.85e113T + 5.06e227T^{2} \) |
| 73 | \( 1 + 4.60e114T + 1.54e229T^{2} \) |
| 79 | \( 1 + 1.49e116T + 2.55e233T^{2} \) |
| 83 | \( 1 + 5.33e116T + 1.11e236T^{2} \) |
| 89 | \( 1 + 7.66e119T + 5.95e239T^{2} \) |
| 97 | \( 1 + 6.12e121T + 2.36e244T^{2} \) |
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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
−13.34650266461559393360168638126, −11.89949971697207658476719269170, −11.10691323183249983626075043219, −8.394958208116487018691608960844, −7.44142548072124501606878173042, −5.93329227200912742038346077400, −4.13369275793966461347509946530, −3.85937191057701662358573665475, −2.71212792918267050983723314463, −1.37802706940567308644698138791,
1.37802706940567308644698138791, 2.71212792918267050983723314463, 3.85937191057701662358573665475, 4.13369275793966461347509946530, 5.93329227200912742038346077400, 7.44142548072124501606878173042, 8.394958208116487018691608960844, 11.10691323183249983626075043219, 11.89949971697207658476719269170, 13.34650266461559393360168638126
