| L(s) = 1 | − 1.28·3-s + 4.24·5-s − 3.37·7-s − 1.34·9-s + 0.495·11-s + 3.53·13-s − 5.46·15-s − 3.96·17-s + 4.34·21-s − 0.943·23-s + 13.0·25-s + 5.59·27-s + 2.66·29-s + 6.83·31-s − 0.638·33-s − 14.3·35-s − 6.73·37-s − 4.54·39-s + 2.42·41-s − 11.9·43-s − 5.69·45-s − 1.11·47-s + 4.37·49-s + 5.10·51-s − 1.01·53-s + 2.10·55-s + 6.59·59-s + ⋯ |
| L(s) = 1 | − 0.743·3-s + 1.89·5-s − 1.27·7-s − 0.447·9-s + 0.149·11-s + 0.979·13-s − 1.41·15-s − 0.962·17-s + 0.947·21-s − 0.196·23-s + 2.60·25-s + 1.07·27-s + 0.495·29-s + 1.22·31-s − 0.111·33-s − 2.41·35-s − 1.10·37-s − 0.728·39-s + 0.378·41-s − 1.82·43-s − 0.849·45-s − 0.163·47-s + 0.624·49-s + 0.715·51-s − 0.139·53-s + 0.283·55-s + 0.858·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.714791024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.714791024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.28T + 3T^{2} \) |
| 5 | \( 1 - 4.24T + 5T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 - 0.495T + 11T^{2} \) |
| 13 | \( 1 - 3.53T + 13T^{2} \) |
| 17 | \( 1 + 3.96T + 17T^{2} \) |
| 23 | \( 1 + 0.943T + 23T^{2} \) |
| 29 | \( 1 - 2.66T + 29T^{2} \) |
| 31 | \( 1 - 6.83T + 31T^{2} \) |
| 37 | \( 1 + 6.73T + 37T^{2} \) |
| 41 | \( 1 - 2.42T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 1.11T + 47T^{2} \) |
| 53 | \( 1 + 1.01T + 53T^{2} \) |
| 59 | \( 1 - 6.59T + 59T^{2} \) |
| 61 | \( 1 - 9.46T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 9.42T + 71T^{2} \) |
| 73 | \( 1 - 3.25T + 73T^{2} \) |
| 79 | \( 1 - 5.75T + 79T^{2} \) |
| 83 | \( 1 - 7.78T + 83T^{2} \) |
| 89 | \( 1 + 2.68T + 89T^{2} \) |
| 97 | \( 1 + 1.43T + 97T^{2} \) |
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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
−8.466275356782814250552016726550, −6.87335499998128307891716416611, −6.41588085272336902063366536102, −6.19117160416482927018473891859, −5.41805231661970723390989070429, −4.76836754424097034415063113520, −3.49853127751309803211625119413, −2.73527426303988249587184237424, −1.86080364842365174152838362813, −0.71473021413484437409781558986,
0.71473021413484437409781558986, 1.86080364842365174152838362813, 2.73527426303988249587184237424, 3.49853127751309803211625119413, 4.76836754424097034415063113520, 5.41805231661970723390989070429, 6.19117160416482927018473891859, 6.41588085272336902063366536102, 6.87335499998128307891716416611, 8.466275356782814250552016726550
