| L(s) = 1 | + 3.16·3-s − 2.23i·5-s + 4.24i·7-s + 7.00·9-s − 7.07i·15-s + 13.4i·21-s + 1.41i·23-s − 5.00·25-s + 12.6·27-s − 8.94i·29-s + 9.48·35-s − 12·41-s + 3.16·43-s − 15.6i·45-s − 9.89i·47-s + ⋯ |
| L(s) = 1 | + 1.82·3-s − 0.999i·5-s + 1.60i·7-s + 2.33·9-s − 1.82i·15-s + 2.92i·21-s + 0.294i·23-s − 1.00·25-s + 2.43·27-s − 1.66i·29-s + 1.60·35-s − 1.87·41-s + 0.482·43-s − 2.33i·45-s − 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.72768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.72768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| good | 3 | \( 1 - 3.16T + 3T^{2} \) |
| 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 8.94iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 + 9.89iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 13.4iT - 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−10.08164302268214710263413629760, −9.408807609495483701642905618522, −8.700271725833440307662848352971, −8.362799426899788378782365167232, −7.42136494465135891279067382449, −5.99284757057185555489680069110, −4.90768817164404374206429114034, −3.80213579155021671167182054169, −2.64000767398197898063795291477, −1.79017758012972534947909954255,
1.64761737012159444220585707173, 3.03181394402838333716211617106, 3.61368859016422403780433774379, 4.59348051411225990931472056424, 6.58619465988684527394949424807, 7.26488381721599218193184842289, 7.82243395380856467937608846883, 8.784042916977487234572735819441, 9.767309156051866700771405142530, 10.38444675993146745884492222213
