L(s) = 1 | + i·2-s + 3-s − 4-s − i·5-s + i·6-s + 4.25·7-s − i·8-s + 9-s + 10-s + 0.913·11-s − 12-s + 2.72i·13-s + 4.25i·14-s − i·15-s + 16-s − 6.25i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s + 1.60·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s + 0.275·11-s − 0.288·12-s + 0.756i·13-s + 1.13i·14-s − 0.258i·15-s + 0.250·16-s − 1.51i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.293904096\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.293904096\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 37 | \( 1 + (-5.43 + 2.72i)T \) |
good | 7 | \( 1 - 4.25T + 7T^{2} \) |
| 11 | \( 1 - 0.913T + 11T^{2} \) |
| 13 | \( 1 - 2.72iT - 13T^{2} \) |
| 17 | \( 1 + 6.25iT - 17T^{2} \) |
| 19 | \( 1 + 4.43iT - 19T^{2} \) |
| 23 | \( 1 - 2.72iT - 23T^{2} \) |
| 29 | \( 1 + 5.34iT - 29T^{2} \) |
| 31 | \( 1 - 8.98iT - 31T^{2} \) |
| 41 | \( 1 + 8.98T + 41T^{2} \) |
| 43 | \( 1 + 3.52iT - 43T^{2} \) |
| 47 | \( 1 - 6.11T + 47T^{2} \) |
| 53 | \( 1 - 6.25T + 53T^{2} \) |
| 59 | \( 1 - 5.45iT - 59T^{2} \) |
| 61 | \( 1 - 10.6iT - 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 7.77T + 73T^{2} \) |
| 79 | \( 1 - 11.9iT - 79T^{2} \) |
| 83 | \( 1 - 0.902T + 83T^{2} \) |
| 89 | \( 1 + 8.72iT - 89T^{2} \) |
| 97 | \( 1 - 12.5iT - 97T^{2} \) |
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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
−9.483770722227138880389997550634, −8.943059200218369812687623247412, −8.290545572158471750615023888431, −7.42266901820572209275231214152, −6.86713850601290828801604571920, −5.42078227766578246088556335888, −4.80357907283861811393399170305, −4.04746385607230922513384260244, −2.48200156597610685879991356126, −1.18847311637681584561643726167,
1.40684933185677598087008931822, 2.25555573660070458621494075485, 3.51886649165010669220867172732, 4.29369474727772272924235365915, 5.32830627209515795677748336589, 6.36125521824055391785509873950, 7.82780420238879658513510495539, 8.055049733986821654611093069506, 8.890204723172767399818380945348, 10.00405483557573174439509922724