| L(s) = 1 | + 0.716·3-s − 3.89·5-s − 0.997·7-s − 2.48·9-s − 5.75·11-s + 4.94·13-s − 2.79·15-s − 7.25·17-s − 7.45·19-s − 0.714·21-s + 3.08·23-s + 10.1·25-s − 3.93·27-s − 9.91·29-s + 5.52·31-s − 4.12·33-s + 3.88·35-s − 6.05·37-s + 3.54·39-s − 4.59·41-s − 2.13·43-s + 9.67·45-s − 3.50·47-s − 6.00·49-s − 5.20·51-s + 8.90·53-s + 22.3·55-s + ⋯ |
| L(s) = 1 | + 0.413·3-s − 1.74·5-s − 0.377·7-s − 0.828·9-s − 1.73·11-s + 1.37·13-s − 0.720·15-s − 1.76·17-s − 1.71·19-s − 0.156·21-s + 0.643·23-s + 2.03·25-s − 0.756·27-s − 1.84·29-s + 0.992·31-s − 0.717·33-s + 0.656·35-s − 0.995·37-s + 0.567·39-s − 0.717·41-s − 0.325·43-s + 1.44·45-s − 0.511·47-s − 0.857·49-s − 0.728·51-s + 1.22·53-s + 3.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1493281820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1493281820\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 0.716T + 3T^{2} \) |
| 5 | \( 1 + 3.89T + 5T^{2} \) |
| 7 | \( 1 + 0.997T + 7T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 + 7.25T + 17T^{2} \) |
| 19 | \( 1 + 7.45T + 19T^{2} \) |
| 23 | \( 1 - 3.08T + 23T^{2} \) |
| 29 | \( 1 + 9.91T + 29T^{2} \) |
| 31 | \( 1 - 5.52T + 31T^{2} \) |
| 37 | \( 1 + 6.05T + 37T^{2} \) |
| 41 | \( 1 + 4.59T + 41T^{2} \) |
| 43 | \( 1 + 2.13T + 43T^{2} \) |
| 47 | \( 1 + 3.50T + 47T^{2} \) |
| 53 | \( 1 - 8.90T + 53T^{2} \) |
| 59 | \( 1 - 4.03T + 59T^{2} \) |
| 61 | \( 1 - 0.824T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 1.36T + 71T^{2} \) |
| 73 | \( 1 + 4.09T + 73T^{2} \) |
| 79 | \( 1 + 8.00T + 79T^{2} \) |
| 83 | \( 1 - 0.372T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.244679980565159816932992693335, −7.51790969960092027409512831365, −6.77636796561563129711872611444, −6.04418418187444041609594177683, −5.05273133720467291529729837947, −4.29758186309246279876120137866, −3.59417020922619540944184700836, −2.95490734516980702879685912044, −2.06867901685743095039868605618, −0.18294007060371458819628579882,
0.18294007060371458819628579882, 2.06867901685743095039868605618, 2.95490734516980702879685912044, 3.59417020922619540944184700836, 4.29758186309246279876120137866, 5.05273133720467291529729837947, 6.04418418187444041609594177683, 6.77636796561563129711872611444, 7.51790969960092027409512831365, 8.244679980565159816932992693335
