| L(s) = 1 | − 4i·3-s + 5i·5-s + 16·7-s + 11·9-s − 60i·11-s − 86i·13-s + 20·15-s + 18·17-s − 44i·19-s − 64i·21-s − 48·23-s − 25·25-s − 152i·27-s + 186i·29-s + 176·31-s + ⋯ |
| L(s) = 1 | − 0.769i·3-s + 0.447i·5-s + 0.863·7-s + 0.407·9-s − 1.64i·11-s − 1.83i·13-s + 0.344·15-s + 0.256·17-s − 0.531i·19-s − 0.665i·21-s − 0.435·23-s − 0.200·25-s − 1.08i·27-s + 1.19i·29-s + 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.211806397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.211806397\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5iT \) |
| good | 3 | \( 1 + 4iT - 27T^{2} \) |
| 7 | \( 1 - 16T + 343T^{2} \) |
| 11 | \( 1 + 60iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 86iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 18T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 48T + 1.21e4T^{2} \) |
| 29 | \( 1 - 186iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 176T + 2.97e4T^{2} \) |
| 37 | \( 1 - 254iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 186T + 6.89e4T^{2} \) |
| 43 | \( 1 + 100iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 168T + 1.03e5T^{2} \) |
| 53 | \( 1 + 498iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 252iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 58iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 1.03e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 168T + 3.57e5T^{2} \) |
| 73 | \( 1 + 506T + 3.89e5T^{2} \) |
| 79 | \( 1 - 272T + 4.93e5T^{2} \) |
| 83 | \( 1 + 948iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 766T + 9.12e5T^{2} \) |
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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
−8.576778381850420114834659477726, −8.189210238505971206402470548770, −7.45514003884469449929755936706, −6.54226808307365916686948466102, −5.72038944332784288825910446124, −4.90343600576618686352199438668, −3.49761377049754483073197938612, −2.73265252885552569789024311973, −1.37191594816374592094971124817, −0.52959907677084989457371569709,
1.46604723324269914626221672084, 2.15655008097861574747557209962, 4.01580766187010385167119342148, 4.41153223044335345375897833048, 5.02882927787274090627974955843, 6.29321664868313514834996007443, 7.26706757995409809430561704735, 7.943905863717243792594823654605, 9.044520360984614485699064851009, 9.600699533506129862482298384315
