| L(s) = 1 | − 20.9i·5-s + (17.6 + 5.46i)7-s + 23.8i·11-s + 74.6i·13-s + 68.6i·17-s + 26.6·19-s − 74.6i·23-s − 313.·25-s − 128.·29-s − 212.·31-s + (114. − 370. i)35-s − 329.·37-s + 182. i·41-s + 260. i·43-s − 401.·47-s + ⋯ |
| L(s) = 1 | − 1.87i·5-s + (0.955 + 0.295i)7-s + 0.653i·11-s + 1.59i·13-s + 0.979i·17-s + 0.321·19-s − 0.677i·23-s − 2.50·25-s − 0.822·29-s − 1.22·31-s + (0.552 − 1.78i)35-s − 1.46·37-s + 0.695i·41-s + 0.925i·43-s − 1.24·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.222 - 0.975i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.222 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9451449793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9451449793\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-17.6 - 5.46i)T \) |
| good | 5 | \( 1 + 20.9iT - 125T^{2} \) |
| 11 | \( 1 - 23.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 74.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 68.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 26.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 74.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 329.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 182. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 260. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 401.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 76.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 901.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 271. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 499. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 299. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 452. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 347. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 775.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 48.7iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3iT - 9.12e5T^{2} \) |
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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
−9.419037484757610259378323839775, −9.055832531528507525459472826067, −8.288683404562811330748208031539, −7.54743761227455997173850165361, −6.30819493615292166744601894532, −5.23418231201669378659970910431, −4.65762547389462118458127835551, −3.94575502958662172389227742693, −1.85319099211412801920272299077, −1.48835524243482334653722756953,
0.21972886846346508358131368985, 1.88812612979730380031905369560, 3.10235580004682165874305488680, 3.59311122062093389255980754618, 5.20956047289701538863624844686, 5.84271679384632808067843186740, 7.09638547305099640601812757068, 7.46752381332326060263424219161, 8.298639252881222695344878470570, 9.523833582821449455843774399023
