L(s) = 1 | + i·3-s − i·5-s − 1.58·7-s − 9-s + 2.94·11-s − 1.23·13-s + 15-s − 1.25i·17-s − 5.26·19-s − 1.58i·21-s + (−1.83 + 4.42i)23-s − 25-s − i·27-s + 7.54·29-s + 0.690i·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 0.598·7-s − 0.333·9-s + 0.887·11-s − 0.341·13-s + 0.258·15-s − 0.304i·17-s − 1.20·19-s − 0.345i·21-s + (−0.383 + 0.923i)23-s − 0.200·25-s − 0.192i·27-s + 1.40·29-s + 0.123i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6607073728\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6607073728\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (1.83 - 4.42i)T \) |
good | 7 | \( 1 + 1.58T + 7T^{2} \) |
| 11 | \( 1 - 2.94T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 + 1.25iT - 17T^{2} \) |
| 19 | \( 1 + 5.26T + 19T^{2} \) |
| 29 | \( 1 - 7.54T + 29T^{2} \) |
| 31 | \( 1 - 0.690iT - 31T^{2} \) |
| 37 | \( 1 - 4.83iT - 37T^{2} \) |
| 41 | \( 1 - 6.98T + 41T^{2} \) |
| 43 | \( 1 + 4.06T + 43T^{2} \) |
| 47 | \( 1 + 8.95iT - 47T^{2} \) |
| 53 | \( 1 + 10.8iT - 53T^{2} \) |
| 59 | \( 1 + 2.33iT - 59T^{2} \) |
| 61 | \( 1 + 5.28iT - 61T^{2} \) |
| 67 | \( 1 - 9.49T + 67T^{2} \) |
| 71 | \( 1 - 13.5iT - 71T^{2} \) |
| 73 | \( 1 + 9.51T + 73T^{2} \) |
| 79 | \( 1 - 3.71T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 + 12.2iT - 89T^{2} \) |
| 97 | \( 1 + 7.70iT - 97T^{2} \) |
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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
−8.265810253140565030240927230213, −7.02603754271619203580517568502, −6.54799097803356455854090653250, −5.74376404536656049379967276613, −4.93189015662513007024782985344, −4.22967124238305021140082244338, −3.55791973598600715399106705249, −2.64007469507417351355136017241, −1.52364791208375558993319913642, −0.18167823485823873528634630326,
1.12705609165544231903942901081, 2.28553674621264413174642475532, 2.91380142300257590455824249728, 3.98357195137896901125080509917, 4.56393271232328804446963742438, 5.82683736449491622359216492216, 6.40509761477078489405812766314, 6.74691051020629719456007648252, 7.62982309312331916117511635744, 8.319793256126636598007523951380