| L(s) = 1 | + (123. + 123. i)7-s + 296. i·11-s + (−16.2 + 16.2i)13-s + (547. − 547. i)17-s + 262. i·19-s + (2.43e3 + 2.43e3i)23-s − 4.32e3·29-s + 8.47e3·31-s + (100. + 100. i)37-s + 1.46e4i·41-s + (9.26e3 − 9.26e3i)43-s + (1.48e3 − 1.48e3i)47-s + 1.39e4i·49-s + (9.26e3 + 9.26e3i)53-s − 2.05e3·59-s + ⋯ |
| L(s) = 1 | + (0.955 + 0.955i)7-s + 0.738i·11-s + (−0.0266 + 0.0266i)13-s + (0.459 − 0.459i)17-s + 0.167i·19-s + (0.959 + 0.959i)23-s − 0.955·29-s + 1.58·31-s + (0.0121 + 0.0121i)37-s + 1.35i·41-s + (0.764 − 0.764i)43-s + (0.0981 − 0.0981i)47-s + 0.827i·49-s + (0.453 + 0.453i)53-s − 0.0767·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0387 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0387 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.508007722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.508007722\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-123. - 123. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 296. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (16.2 - 16.2i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-547. + 547. i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 262. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-2.43e3 - 2.43e3i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 4.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-100. - 100. i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.46e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-9.26e3 + 9.26e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (-1.48e3 + 1.48e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (-9.26e3 - 9.26e3i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 2.05e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.08e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (9.22e3 + 9.22e3i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 1.05e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (2.04e4 - 2.04e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 + 5.88e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (6.18e4 + 6.18e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 4.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-7.54e4 - 7.54e4i)T + 8.58e9iT^{2} \) |
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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
−9.487667468119255176093103179763, −8.793664632494056318074764946611, −7.86645955187851209313529371174, −7.20473785070336627485886663846, −5.99407064988405788622482928669, −5.18564189314990991656405921971, −4.45492724535106846056058461468, −3.09531646745968871807259963521, −2.09420739443404372974480796416, −1.12190681133577770289727877167,
0.54344674892668413947625007597, 1.35458517353053972295565716163, 2.69159254988599277846124944502, 3.85111613102911181490637010911, 4.66089155043110934482007234518, 5.62256556280183084776251749901, 6.65270030844275127900229397025, 7.55477576289196110511935721305, 8.248905015718795318139446307967, 9.033864847080424723062058907315
