| L(s) = 1 | − 3i·3-s + (10 − 5i)5-s − 10i·7-s − 9·9-s + 46·11-s + 34i·13-s + (−15 − 30i)15-s − 66i·17-s + 104·19-s − 30·21-s + 164i·23-s + (75 − 100i)25-s + 27i·27-s + 224·29-s − 72·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.894 − 0.447i)5-s − 0.539i·7-s − 0.333·9-s + 1.26·11-s + 0.725i·13-s + (−0.258 − 0.516i)15-s − 0.941i·17-s + 1.25·19-s − 0.311·21-s + 1.48i·23-s + (0.599 − 0.800i)25-s + 0.192i·27-s + 1.43·29-s − 0.417·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.923229534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.923229534\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 + (-10 + 5i)T \) |
| good | 7 | \( 1 + 10iT - 343T^{2} \) |
| 11 | \( 1 - 46T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 66iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 104T + 6.85e3T^{2} \) |
| 23 | \( 1 - 164iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 224T + 2.43e4T^{2} \) |
| 31 | \( 1 + 72T + 2.97e4T^{2} \) |
| 37 | \( 1 + 22iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 194T + 6.89e4T^{2} \) |
| 43 | \( 1 + 108iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 480iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 286iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 426T + 2.05e5T^{2} \) |
| 61 | \( 1 + 698T + 2.26e5T^{2} \) |
| 67 | \( 1 - 328iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 188T + 3.57e5T^{2} \) |
| 73 | \( 1 + 740iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 412iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3iT - 9.12e5T^{2} \) |
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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
−9.359856705317194039034172823374, −8.929821366979139158928071203640, −7.62817311571429252907055490275, −6.96046497984652316455803950632, −6.13924826458358244956256096774, −5.22474665674629472980698275071, −4.19129528310316115391884909280, −2.94929778838952601883422259703, −1.59800947319570399613448394829, −0.923918485388704903815822943164,
1.10666592587872645003552225731, 2.44913665195479619226571796935, 3.37420412480245023801702835326, 4.50991321428618044809814007283, 5.60250513752206878391347172866, 6.19671970509778178056173638606, 7.09181458801760521619867815219, 8.420437932161266074418861757186, 8.998086635960919481911835006963, 9.902919696566186985700016688188
