L(s) = 1 | − 2i·2-s − 4·4-s + 22i·7-s + 8i·8-s − 12·11-s + 38i·13-s + 44·14-s + 16·16-s + 105i·17-s + 157·19-s + 24i·22-s − 117i·23-s + 76·26-s − 88i·28-s − 66·29-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.18i·7-s + 0.353i·8-s − 0.328·11-s + 0.810i·13-s + 0.839·14-s + 0.250·16-s + 1.49i·17-s + 1.89·19-s + 0.232i·22-s − 1.06i·23-s + 0.573·26-s − 0.593i·28-s − 0.422·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9825646034\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9825646034\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 22iT - 343T^{2} \) |
| 11 | \( 1 + 12T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 105iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 157T + 6.85e3T^{2} \) |
| 23 | \( 1 + 117iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 66T + 2.43e4T^{2} \) |
| 31 | \( 1 + 25T + 2.97e4T^{2} \) |
| 37 | \( 1 + 314iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 504T + 6.89e4T^{2} \) |
| 43 | \( 1 - 380iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 252iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 3iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 318T + 2.05e5T^{2} \) |
| 61 | \( 1 - 293T + 2.26e5T^{2} \) |
| 67 | \( 1 - 322iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 120T + 3.57e5T^{2} \) |
| 73 | \( 1 - 44iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 917T + 4.93e5T^{2} \) |
| 83 | \( 1 - 309iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.27e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.32e3iT - 9.12e5T^{2} \) |
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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
−9.547309246470279361367649356665, −8.774161673056846870998681420824, −8.181707610513285595457663948353, −7.07933726437958444918465428674, −5.96418774353224261746537223386, −5.33029618947511257544340545158, −4.28655536222325652550279685306, −3.25996529209873327360309095430, −2.31815909939069258153273437822, −1.38319702431786030342843126942,
0.24331252468986436478893736897, 1.22315563001457299801655859828, 3.03730823304192081714980406823, 3.76935632079978045677505634475, 5.13176057874360848695290270392, 5.35507128140237101930150216017, 6.80127324522516639880146112948, 7.34644571316452324760322061060, 7.86477713327128268962068854360, 8.908779307400659211550841679580