| L(s) = 1 | − i·2-s − 4-s + i·5-s + i·8-s + 10-s + (−1 − 3.46i)13-s + 16-s − i·20-s + 6.92·23-s − 25-s + (−3.46 + i)26-s + 6.92·29-s − 6.92i·31-s − i·32-s − 6.92i·37-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.447i·5-s + 0.353i·8-s + 0.316·10-s + (−0.277 − 0.960i)13-s + 0.250·16-s − 0.223i·20-s + 1.44·23-s − 0.200·25-s + (−0.679 + 0.196i)26-s + 1.28·29-s − 1.24i·31-s − 0.176i·32-s − 1.13i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.482025907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.482025907\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 6.92iT - 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 6.92iT - 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.699620396131745592290520132501, −8.989417266437124296340656467072, −7.999367137372579090556011938334, −7.26445854273556334909701564984, −6.16255842665651098379277581075, −5.24751364835872490553065116468, −4.26298730135295017975910021752, −3.15760663608124640694741079619, −2.40029253587341501132403950257, −0.792314659587976117963536040617,
1.15248015792437480460930167456, 2.76410850700942663265718571614, 4.11715947777933840013543102997, 4.87043443134190095475005256719, 5.70776791511713491441687518610, 6.82377613185500686632308489076, 7.23888546298295147466743931275, 8.534052897301388180923807231195, 8.830121092550688019444339660147, 9.783007986065791989733124743382
