L(s) = 1 | + 2.41i·3-s − 0.828i·7-s − 2.82·9-s − 5.24·11-s − 5.65i·13-s − 0.171i·17-s + 1.58·19-s + 1.99·21-s − 4.82i·23-s + 0.414i·27-s + 8·29-s + 0.828·31-s − 12.6i·33-s + 7.65i·37-s + 13.6·39-s + ⋯ |
L(s) = 1 | + 1.39i·3-s − 0.313i·7-s − 0.942·9-s − 1.58·11-s − 1.56i·13-s − 0.0416i·17-s + 0.363·19-s + 0.436·21-s − 1.00i·23-s + 0.0797i·27-s + 1.48·29-s + 0.148·31-s − 2.20i·33-s + 1.25i·37-s + 2.18·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.573561096\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.573561096\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 2.41iT - 3T^{2} \) |
| 7 | \( 1 + 0.828iT - 7T^{2} \) |
| 11 | \( 1 + 5.24T + 11T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 0.171iT - 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 23 | \( 1 + 4.82iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 0.828T + 31T^{2} \) |
| 37 | \( 1 - 7.65iT - 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 9.65iT - 47T^{2} \) |
| 53 | \( 1 - 7.65iT - 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 2.75iT - 67T^{2} \) |
| 71 | \( 1 - 9.65T + 71T^{2} \) |
| 73 | \( 1 + 5.82iT - 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 1.24iT - 83T^{2} \) |
| 89 | \( 1 + 6.17T + 89T^{2} \) |
| 97 | \( 1 + 17.3iT - 97T^{2} \) |
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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
−8.864397634245682150303722798086, −7.980328954507207595381959617025, −7.66916444501208227545627127408, −6.32595186564942816715541268597, −5.56132818329605838215511769342, −4.78945963628878229836538094404, −4.37796570950588149668113930921, −2.97135219300863248273948531392, −2.87260710518587221396619414197, −0.77955967243387585716311151967,
0.70762900335616452250646992932, 2.02698896702243112711522493667, 2.43362422814152774647991266828, 3.70228757176455576848050922442, 4.85755038359512127093859230172, 5.63087714976238310658672095417, 6.40118596675179491683494071484, 7.19283281870393667078837072609, 7.59438604661351827332896620175, 8.399205988583068362521587816140