| L(s) = 1 | + i·5-s + 1.41i·7-s − 3.55·11-s − 6.57·13-s + 0.138i·17-s − 4.96i·19-s + 3.16·23-s − 25-s + 2.19i·29-s + 9.16i·31-s − 1.41·35-s + 0.723·37-s − 1.21i·41-s + 7.30i·43-s + 0.494·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 0.534i·7-s − 1.07·11-s − 1.82·13-s + 0.0335i·17-s − 1.13i·19-s + 0.659·23-s − 0.200·25-s + 0.407i·29-s + 1.64i·31-s − 0.239·35-s + 0.119·37-s − 0.190i·41-s + 1.11i·43-s + 0.0721·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9234212879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9234212879\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 3.55T + 11T^{2} \) |
| 13 | \( 1 + 6.57T + 13T^{2} \) |
| 17 | \( 1 - 0.138iT - 17T^{2} \) |
| 19 | \( 1 + 4.96iT - 19T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 - 2.19iT - 29T^{2} \) |
| 31 | \( 1 - 9.16iT - 31T^{2} \) |
| 37 | \( 1 - 0.723T + 37T^{2} \) |
| 41 | \( 1 + 1.21iT - 41T^{2} \) |
| 43 | \( 1 - 7.30iT - 43T^{2} \) |
| 47 | \( 1 - 0.494T + 47T^{2} \) |
| 53 | \( 1 + 12.6iT - 53T^{2} \) |
| 59 | \( 1 + 6.29T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 13.8iT - 67T^{2} \) |
| 71 | \( 1 + 8.47T + 71T^{2} \) |
| 73 | \( 1 + 5.10T + 73T^{2} \) |
| 79 | \( 1 + 6.81iT - 79T^{2} \) |
| 83 | \( 1 + 4.48T + 83T^{2} \) |
| 89 | \( 1 + 11.2iT - 89T^{2} \) |
| 97 | \( 1 - 9.30T + 97T^{2} \) |
| 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
−7.935041674956165888590019859646, −7.17097021436274615329483027325, −6.84750744378620815529412496540, −5.75854805879868079226543984738, −4.97973833129552215924561909224, −4.69925944039843637634902398108, −3.19323913800314850369120655051, −2.75252893529813862300864346673, −1.94519060470330701265896931466, −0.30113037241840129123101839012,
0.792784604787799776695625534659, 2.14489580923368089287263154493, 2.77376550792941511019264777793, 3.96155040485297058483792551393, 4.56774674024260410113183399633, 5.37779094754097360620151766077, 5.88036759962422843625454878364, 7.07549717177254816322842629047, 7.53177003045404514727312473986, 8.038990557497023422681223341026
