| L(s) = 1 | − 3·3-s − 15.3·5-s − 23.4·7-s + 9·9-s + 0.201·11-s − 21.5·13-s + 45.9·15-s − 117.·17-s − 143.·19-s + 70.3·21-s + 23·23-s + 109.·25-s − 27·27-s − 202.·29-s − 21.7·31-s − 0.603·33-s + 359.·35-s + 62.3·37-s + 64.7·39-s + 230.·41-s + 282.·43-s − 137.·45-s + 329.·47-s + 207.·49-s + 353.·51-s + 353.·53-s − 3.07·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.36·5-s − 1.26·7-s + 0.333·9-s + 0.00550·11-s − 0.460·13-s + 0.790·15-s − 1.67·17-s − 1.72·19-s + 0.731·21-s + 0.208·23-s + 0.876·25-s − 0.192·27-s − 1.29·29-s − 0.126·31-s − 0.00318·33-s + 1.73·35-s + 0.277·37-s + 0.266·39-s + 0.879·41-s + 1.00·43-s − 0.456·45-s + 1.02·47-s + 0.604·49-s + 0.969·51-s + 0.915·53-s − 0.00754·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2300025504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2300025504\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 + 15.3T + 125T^{2} \) |
| 7 | \( 1 + 23.4T + 343T^{2} \) |
| 11 | \( 1 - 0.201T + 1.33e3T^{2} \) |
| 13 | \( 1 + 21.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 143.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 202.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 21.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 62.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 230.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 282.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 329.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 353.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 274.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 48.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 518.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 959.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 887.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 22.6T + 4.93e5T^{2} \) |
| 83 | \( 1 - 297.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 926.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 482.T + 9.12e5T^{2} \) |
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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
−10.68831662653490366372437614285, −9.439676352284864840604818810118, −8.679654681053926374106984607093, −7.49477767896691336373836698121, −6.79062377395671677757698042800, −5.93958329822060651267944572901, −4.44439475525354240040167091830, −3.88199459051288060782232585828, −2.45980045672589764985553401458, −0.27931873183518841282782309369,
0.27931873183518841282782309369, 2.45980045672589764985553401458, 3.88199459051288060782232585828, 4.44439475525354240040167091830, 5.93958329822060651267944572901, 6.79062377395671677757698042800, 7.49477767896691336373836698121, 8.679654681053926374106984607093, 9.439676352284864840604818810118, 10.68831662653490366372437614285
