L(s) = 1 | + 2.36i·3-s + i·5-s + 4.10·7-s − 2.61·9-s + 1.58·11-s + 1.61·13-s − 2.36·15-s + 0.614i·17-s + 3.28·19-s + 9.71i·21-s + (2.21 − 4.25i)23-s − 25-s + 0.914i·27-s + 9.48·29-s − 3.91i·31-s + ⋯ |
L(s) = 1 | + 1.36i·3-s + 0.447i·5-s + 1.55·7-s − 0.871·9-s + 0.477·11-s + 0.447·13-s − 0.611·15-s + 0.148i·17-s + 0.753·19-s + 2.12i·21-s + (0.462 − 0.886i)23-s − 0.200·25-s + 0.175i·27-s + 1.76·29-s − 0.702i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0423 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0423 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.346109364\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.346109364\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-2.21 + 4.25i)T \) |
good | 3 | \( 1 - 2.36iT - 3T^{2} \) |
| 7 | \( 1 - 4.10T + 7T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 - 0.614iT - 17T^{2} \) |
| 19 | \( 1 - 3.28T + 19T^{2} \) |
| 29 | \( 1 - 9.48T + 29T^{2} \) |
| 31 | \( 1 + 3.91iT - 31T^{2} \) |
| 37 | \( 1 - 3.87iT - 37T^{2} \) |
| 41 | \( 1 - 3.45T + 41T^{2} \) |
| 43 | \( 1 + 8.20T + 43T^{2} \) |
| 47 | \( 1 - 0.487iT - 47T^{2} \) |
| 53 | \( 1 + 0.645iT - 53T^{2} \) |
| 59 | \( 1 + 14.3iT - 59T^{2} \) |
| 61 | \( 1 - 6.94iT - 61T^{2} \) |
| 67 | \( 1 - 9.87T + 67T^{2} \) |
| 71 | \( 1 + 9.75iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 9.87T + 83T^{2} \) |
| 89 | \( 1 + 5.22iT - 89T^{2} \) |
| 97 | \( 1 - 12.6iT - 97T^{2} \) |
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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
−9.577483346243847426882844460734, −8.613706473163892043564577634660, −8.184671858999712540138106726814, −7.10231030937664624441402932981, −6.13947635445469321187231451861, −5.06480213812037140612002220849, −4.60531356749764267268526494442, −3.77427669148454846294173192919, −2.75322238012677529721126416119, −1.34583928775719668556693600370,
1.14195735334949791121489755555, 1.51594721080367911232351218737, 2.78515393975727289507656386474, 4.18533954368390298513654642124, 5.09030804469229739913251240165, 5.84787521006686997062251469994, 6.91836155288824642582026675604, 7.42041835469239981131497711915, 8.331846261581552408870864106398, 8.596704686983209283088287425367