| L(s) = 1 | − 0.317i·5-s + 1.44i·7-s − 1.09·11-s − 2.89·13-s − 3.46i·17-s − 4.89i·19-s − 2.82·23-s + 4.89·25-s − 9.12i·29-s + 7.44i·31-s + 0.460·35-s − 4.89·37-s − 9.75i·41-s − 6.89i·43-s + 9.12·47-s + ⋯ |
| L(s) = 1 | − 0.142i·5-s + 0.547i·7-s − 0.330·11-s − 0.804·13-s − 0.840i·17-s − 1.12i·19-s − 0.589·23-s + 0.979·25-s − 1.69i·29-s + 1.33i·31-s + 0.0778·35-s − 0.805·37-s − 1.52i·41-s − 1.05i·43-s + 1.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.132317274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.132317274\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.317iT - 5T^{2} \) |
| 7 | \( 1 - 1.44iT - 7T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 9.12iT - 29T^{2} \) |
| 31 | \( 1 - 7.44iT - 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 + 9.75iT - 41T^{2} \) |
| 43 | \( 1 + 6.89iT - 43T^{2} \) |
| 47 | \( 1 - 9.12T + 47T^{2} \) |
| 53 | \( 1 + 4.41iT - 53T^{2} \) |
| 59 | \( 1 - 9.12T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 5.10iT - 67T^{2} \) |
| 71 | \( 1 + 7.56T + 71T^{2} \) |
| 73 | \( 1 - 1.89T + 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 11.9iT - 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−8.951180014244978029454962154394, −8.583574980380660541785318548491, −7.34552493576272980753780507308, −6.95825516960830509496074052035, −5.70337313582588269662817226660, −5.10681748306194925363438720781, −4.21626097446394384696673098653, −2.90380233151201926832612974100, −2.18495096536102471051271913203, −0.43539434004673015020903518404,
1.33382656468913062578702097741, 2.58924248601055898056468370426, 3.65989893594197516045266365702, 4.50846476051776193997607470620, 5.48336746160743229037931100469, 6.32803217519549978561002336747, 7.23614385977551999209296479794, 7.86607364430354298873864302816, 8.665040697364095451587694992666, 9.628273687320824943031719557720
