L(s) = 1 | + (−0.980 − 0.195i)2-s + (0.275 + 0.275i)3-s + (0.923 + 0.382i)4-s + (0.707 − 0.707i)5-s + (−0.216 − 0.324i)6-s + (−0.831 − 0.555i)8-s − 0.847i·9-s + (−0.831 + 0.555i)10-s + (−0.541 + 0.541i)11-s + (0.149 + 0.360i)12-s + (0.785 + 0.785i)13-s + 0.390·15-s + (0.707 + 0.707i)16-s + (−0.165 + 0.831i)18-s + (0.707 + 0.707i)19-s + (0.923 − 0.382i)20-s + ⋯ |
L(s) = 1 | + (−0.980 − 0.195i)2-s + (0.275 + 0.275i)3-s + (0.923 + 0.382i)4-s + (0.707 − 0.707i)5-s + (−0.216 − 0.324i)6-s + (−0.831 − 0.555i)8-s − 0.847i·9-s + (−0.831 + 0.555i)10-s + (−0.541 + 0.541i)11-s + (0.149 + 0.360i)12-s + (0.785 + 0.785i)13-s + 0.390·15-s + (0.707 + 0.707i)16-s + (−0.165 + 0.831i)18-s + (0.707 + 0.707i)19-s + (0.923 − 0.382i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9199584875\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9199584875\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.980 + 0.195i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 19 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-0.275 - 0.275i)T + iT^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 13 | \( 1 + (-0.785 - 0.785i)T + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1.38 + 1.38i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.275 - 0.275i)T - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 67 | \( 1 + (1.17 + 1.17i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + 1.11T + T^{2} \) |
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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
−9.509779784053368896543290155404, −9.032387428836150261172541093916, −8.256970750859021669266161021550, −7.41432917541547060339547803458, −6.38131073267588228343635247619, −5.76408114663102189835350435546, −4.44825387932936429858409441047, −3.44866343149143682165035361677, −2.25603028680066625741206286979, −1.19833927235473156341067808003,
1.35704140994080236832109299407, 2.59259490689797777190493511251, 3.14319568199063472739406522516, 5.07572483897199131807252905547, 5.84964008968626169034411544352, 6.60356660641400198848460727457, 7.51099965625427066454983061457, 8.085818328998989269941273502370, 8.812420463493615495944967425649, 9.790019926893020149310404019874