| L(s) = 1 | + i·7-s + 3·9-s − 5·11-s + 6i·13-s + 4i·17-s + 6·19-s − 3i·23-s + 3·29-s + 2·31-s + 7i·37-s − 4·41-s + 7i·43-s − 2i·47-s − 49-s + 10i·53-s + ⋯ |
| L(s) = 1 | + 0.377i·7-s + 9-s − 1.50·11-s + 1.66i·13-s + 0.970i·17-s + 1.37·19-s − 0.625i·23-s + 0.557·29-s + 0.359·31-s + 1.15i·37-s − 0.624·41-s + 1.06i·43-s − 0.291i·47-s − 0.142·49-s + 1.37i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20147 + 0.742553i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20147 + 0.742553i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 7iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 13T + 71T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−10.39630201846709653688252958820, −9.898435165927122424821339861481, −8.880575765166552278343172025796, −7.986143326869911592311018799088, −7.11519169640077653996075052143, −6.22512540844802711355733326436, −5.05039575756716157407313005772, −4.26109177807554697867478453567, −2.86740551482613276423128265527, −1.60049408733384802633414554803,
0.790133013805134540606127223683, 2.60098483233759492199007751435, 3.61816846242806993735719031260, 5.06162779077396476297767527010, 5.49812720141374618627204639874, 7.10228689012684436915297422111, 7.56363556880996454325480902808, 8.396994660749056155037365720496, 9.833678509456435057677641150799, 10.10741351579027436891973055389
