| L(s) = 1 | + 7.11·3-s − 5·5-s − 13.7·7-s + 23.5·9-s − 24.2·11-s + 3.05·13-s − 35.5·15-s + 63.1·17-s + 2.07·19-s − 98.0·21-s + 23·23-s + 25·25-s − 24.1·27-s − 8.16·29-s + 156.·31-s − 172.·33-s + 68.9·35-s + 302.·37-s + 21.6·39-s − 42.7·41-s − 215.·43-s − 117.·45-s − 247.·47-s − 152.·49-s + 448.·51-s + 600.·53-s + 121.·55-s + ⋯ |
| L(s) = 1 | + 1.36·3-s − 0.447·5-s − 0.744·7-s + 0.874·9-s − 0.664·11-s + 0.0650·13-s − 0.612·15-s + 0.900·17-s + 0.0250·19-s − 1.01·21-s + 0.208·23-s + 0.200·25-s − 0.172·27-s − 0.0522·29-s + 0.909·31-s − 0.909·33-s + 0.332·35-s + 1.34·37-s + 0.0890·39-s − 0.162·41-s − 0.763·43-s − 0.390·45-s − 0.766·47-s − 0.445·49-s + 1.23·51-s + 1.55·53-s + 0.297·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.838916246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.838916246\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 7.11T + 27T^{2} \) |
| 7 | \( 1 + 13.7T + 343T^{2} \) |
| 11 | \( 1 + 24.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.05T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.07T + 6.85e3T^{2} \) |
| 29 | \( 1 + 8.16T + 2.43e4T^{2} \) |
| 31 | \( 1 - 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 302.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 42.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 215.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 247.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 600.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 92.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 532.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 30.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 736.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 349.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 301.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 139.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 859.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 927.T + 9.12e5T^{2} \) |
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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
−8.728916669417350986315264685635, −8.159395073066613159400846394028, −7.56459683228581491622589014341, −6.70987884272629176473581934785, −5.68991880655154302039554008333, −4.60568040043981020987584230720, −3.55319055656061468288156224309, −3.06029481392907914397393668095, −2.16888478504306093675206984303, −0.72124408796283364923421036677,
0.72124408796283364923421036677, 2.16888478504306093675206984303, 3.06029481392907914397393668095, 3.55319055656061468288156224309, 4.60568040043981020987584230720, 5.68991880655154302039554008333, 6.70987884272629176473581934785, 7.56459683228581491622589014341, 8.159395073066613159400846394028, 8.728916669417350986315264685635
