| L(s) = 1 | − i·5-s + 4.24·7-s + 5.65i·11-s + 2i·13-s − 6·17-s − 2.82i·19-s + 7.07·23-s − 25-s − 4i·29-s + 2.82·31-s − 4.24i·35-s + 2i·37-s + 8·41-s + 1.41i·43-s − 1.41·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 1.60·7-s + 1.70i·11-s + 0.554i·13-s − 1.45·17-s − 0.648i·19-s + 1.47·23-s − 0.200·25-s − 0.742i·29-s + 0.508·31-s − 0.717i·35-s + 0.328i·37-s + 1.24·41-s + 0.215i·43-s − 0.206·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.340484013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.340484013\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 1.41iT - 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 2.82iT - 59T^{2} \) |
| 61 | \( 1 - 14iT - 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 12.7iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.277594297057721337843791739189, −7.30931991891321083696322468182, −7.11041797535423060577724005022, −6.05005893903500241344736306431, −4.98318875561479592609496373966, −4.55877529660135868285585883638, −4.26710657276399351600561317578, −2.59409602731575447451225997185, −1.97600694214068064526719891345, −1.09372770515835110352837911073,
0.67146474165776443736312180241, 1.71113141720879485598061725570, 2.72493501628405493348547018346, 3.48445117736586501386896920363, 4.46176179718632249754436629867, 5.14465781847026199350209162475, 5.84694453977664795763714933248, 6.57768628598532614325953541185, 7.44903360285837166540585977904, 8.060176016067707635961384901130
