| L(s) = 1 | + i·3-s + 2·9-s − 11-s − 5i·13-s − i·17-s + 6·19-s − 4i·23-s + 5i·27-s − 3·29-s + 2·31-s − i·33-s − 8i·37-s + 5·39-s − 10·41-s − 2i·43-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.666·9-s − 0.301·11-s − 1.38i·13-s − 0.242i·17-s + 1.37·19-s − 0.834i·23-s + 0.962i·27-s − 0.557·29-s + 0.359·31-s − 0.174i·33-s − 1.31i·37-s + 0.800·39-s − 1.56·41-s − 0.304i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.638038455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.638038455\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 - 3iT - 97T^{2} \) |
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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
−7.902442672220571329731694647512, −7.64142340022396011646497228486, −6.72826854177635356416922438819, −5.80436373880083372645068435375, −5.12049585700382583389213360473, −4.53994739523548076393664159338, −3.47687900923214761571225487748, −2.97091408801980420023771552852, −1.69151683799610837406794011830, −0.45940019415673966553784621328,
1.24136756422567987164184934698, 1.84540430148615545707932662814, 3.00064520257618873930694319209, 3.89398435673951544669968263752, 4.71891267453636074536423633706, 5.47331316701825921266528869724, 6.43410110139514074923901713489, 6.96801728605908224285553901697, 7.57871120688647191177386278863, 8.242135337932297305403067708443
