| L(s) = 1 | + 3-s − 3i·5-s + i·7-s + 9-s + 5i·11-s + (−3 − 2i)13-s − 3i·15-s − 7·17-s + 3i·19-s + i·21-s + 3·23-s − 4·25-s + 27-s + 7·29-s − 4i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34i·5-s + 0.377i·7-s + 0.333·9-s + 1.50i·11-s + (−0.832 − 0.554i)13-s − 0.774i·15-s − 1.69·17-s + 0.688i·19-s + 0.218i·21-s + 0.625·23-s − 0.800·25-s + 0.192·27-s + 1.29·29-s − 0.718i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.724072861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724072861\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 5 | \( 1 + 3iT - 5T^{2} \) |
| 11 | \( 1 - 5iT - 11T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 - 3iT - 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 - 7T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 - 11iT - 37T^{2} \) |
| 41 | \( 1 - 4iT - 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 - iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.544044151320316744669069515800, −7.923303115641193869754557193062, −7.12937002305185184112781904543, −6.37878365873073951623957656497, −5.26555711429031467626671167428, −4.60133637342754384402343570425, −4.30936913283081930403085468723, −2.84404030646306228087673867861, −2.13718862296991686395523833174, −1.13527797209412091547493601171,
0.46029575132970610110525787053, 2.15430324263317585182788741404, 2.75431381074693281224859169269, 3.51209657718243667605373702233, 4.31863439684940866146042430018, 5.27214938639184052746118228575, 6.37446540489992413742160049913, 6.92378120904244843419749044823, 7.25606675196069157271812667391, 8.424819956515758711719724765193
