| L(s) = 1 | + i·3-s − 1.41·5-s + 2.82i·7-s − 9-s + 5.09·11-s + 3.60i·13-s − 1.41i·15-s + 2·17-s − 2.82·21-s − 2.99·25-s − i·27-s + 7.21i·29-s + 5.09i·33-s − 4.00i·35-s + 8.48·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.632·5-s + 1.06i·7-s − 0.333·9-s + 1.53·11-s + 0.999i·13-s − 0.365i·15-s + 0.485·17-s − 0.617·21-s − 0.599·25-s − 0.192i·27-s + 1.33i·29-s + 0.887i·33-s − 0.676i·35-s + 1.39·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.437232906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.437232906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 13 | \( 1 - 3.60iT \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 5.09T + 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 7.21iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 5.09iT - 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 7.07iT - 47T^{2} \) |
| 53 | \( 1 - 7.21iT - 53T^{2} \) |
| 59 | \( 1 - 5.09T + 59T^{2} \) |
| 61 | \( 1 - 7.21iT - 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 4.24iT - 71T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 + 7.21T + 79T^{2} \) |
| 83 | \( 1 + 5.09T + 83T^{2} \) |
| 89 | \( 1 - 5.09iT - 89T^{2} \) |
| 97 | \( 1 - 10.1iT - 97T^{2} \) |
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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
−9.031521520912237642981488679768, −8.795677813385839265024671831922, −7.73619512582706971943696397896, −6.84755861712196036303149335183, −6.08322129726751444327486207667, −5.28304524385967002021924018478, −4.23788276960488401417427765044, −3.75957731052884989160607280629, −2.64674413975614802575074856484, −1.42229602980889664904424380952,
0.53118997285082643911991208863, 1.43137462873737540768092231883, 2.90174195299476151557331431197, 3.87574561306896422944909153887, 4.37101820821259432821524366130, 5.70244320653281867973740604183, 6.41457193936221230491544604641, 7.20952603156384838982336078329, 7.83889888561318113982702628243, 8.341706859441301297011520296474
