| L(s) = 1 | + (1.08 + 1.95i)5-s + (0.595 − 2.57i)7-s − 3.74i·11-s + 3.36·13-s + 0.841i·17-s − 5.59i·19-s − 2.35·23-s + (−2.64 + 4.24i)25-s + 1.41i·29-s − 8.66i·31-s + (5.68 − 1.63i)35-s − 5.15i·37-s + 5.74·41-s + 3.32i·43-s + 6.43i·47-s + ⋯ |
| L(s) = 1 | + (0.485 + 0.874i)5-s + (0.224 − 0.974i)7-s − 1.12i·11-s + 0.931·13-s + 0.204i·17-s − 1.28i·19-s − 0.490·23-s + (−0.529 + 0.848i)25-s + 0.262i·29-s − 1.55i·31-s + (0.961 − 0.276i)35-s − 0.847i·37-s + 0.896·41-s + 0.507i·43-s + 0.938i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.840721322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.840721322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.08 - 1.95i)T \) |
| 7 | \( 1 + (-0.595 + 2.57i)T \) |
| good | 11 | \( 1 + 3.74iT - 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 - 0.841iT - 17T^{2} \) |
| 19 | \( 1 + 5.59iT - 19T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 8.66iT - 31T^{2} \) |
| 37 | \( 1 + 5.15iT - 37T^{2} \) |
| 41 | \( 1 - 5.74T + 41T^{2} \) |
| 43 | \( 1 - 3.32iT - 43T^{2} \) |
| 47 | \( 1 - 6.43iT - 47T^{2} \) |
| 53 | \( 1 - 9.64T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 1.82iT - 67T^{2} \) |
| 71 | \( 1 + 3.74iT - 71T^{2} \) |
| 73 | \( 1 + 0.979T + 73T^{2} \) |
| 79 | \( 1 + 6.58T + 79T^{2} \) |
| 83 | \( 1 - 12.5iT - 83T^{2} \) |
| 89 | \( 1 + 2.16T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.675683300362167465983734093699, −8.813211344156084139737896458274, −7.893912790214104478148885922304, −7.10808070739035233509534168058, −6.26503743033129661880223264519, −5.62339307092892661474019398681, −4.24401091321916350830215258510, −3.46169361443652963394168300138, −2.36900621091324165336763059437, −0.853653300453039662919897468046,
1.41915419380110760244835623013, 2.29089283982403971794506196687, 3.76449862467358939965175734071, 4.79063407599028087901721877816, 5.56150676654092587406447831371, 6.25731449893128920601172811712, 7.38840756516812837015163482000, 8.506510250955337480879534154832, 8.732349366017579864408679932286, 9.846314623709979854880838087326
