| L(s) = 1 | + i·2-s + i·3-s − 4-s + 1.78i·5-s − 6-s − i·8-s − 9-s − 1.78·10-s + (−3.15 + 1.02i)11-s − i·12-s + 6.37·13-s − 1.78·15-s + 16-s − 0.106·17-s − i·18-s + 4.15·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.799i·5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s − 0.565·10-s + (−0.951 + 0.308i)11-s − 0.288i·12-s + 1.76·13-s − 0.461·15-s + 0.250·16-s − 0.0257·17-s − 0.235i·18-s + 0.952·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.806020105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.806020105\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (3.15 - 1.02i)T \) |
| good | 5 | \( 1 - 1.78iT - 5T^{2} \) |
| 13 | \( 1 - 6.37T + 13T^{2} \) |
| 17 | \( 1 + 0.106T + 17T^{2} \) |
| 19 | \( 1 - 4.15T + 19T^{2} \) |
| 23 | \( 1 - 7.95T + 23T^{2} \) |
| 29 | \( 1 - 7.65iT - 29T^{2} \) |
| 31 | \( 1 + 1.27iT - 31T^{2} \) |
| 37 | \( 1 - 4.52T + 37T^{2} \) |
| 41 | \( 1 - 0.0321T + 41T^{2} \) |
| 43 | \( 1 - 6.87iT - 43T^{2} \) |
| 47 | \( 1 - 9.88iT - 47T^{2} \) |
| 53 | \( 1 + 0.627T + 53T^{2} \) |
| 59 | \( 1 + 12.4iT - 59T^{2} \) |
| 61 | \( 1 + 9.95T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 8.42T + 71T^{2} \) |
| 73 | \( 1 + 0.125T + 73T^{2} \) |
| 79 | \( 1 + 10.1iT - 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 5.36iT - 89T^{2} \) |
| 97 | \( 1 - 15.7iT - 97T^{2} \) |
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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
−8.975183294386491895133311144604, −8.171507932212648782345002039891, −7.46581331163676822101264913173, −6.71200285869713581153622549013, −6.01431829495235850663576538946, −5.20799012689616770462473195225, −4.52564735545922135027056721899, −3.33451134193957439118436586054, −2.95960365522512847718550424336, −1.20179162467525464734821760701,
0.68657432964413323449201382952, 1.35851482365329429954883293635, 2.61375756458343069934816326124, 3.39701931303092346580491372184, 4.38257629287042526578061246541, 5.34313486446379547844607128794, 5.81524651390622617150663363037, 6.90050859511616764643429278658, 7.77284826148536826144826806376, 8.579560589765641595690032137416
