| L(s) = 1 | − 1.41i·2-s + 3.16·3-s − 2.00·4-s − 4.47i·6-s + (1.58 + 2.12i)7-s + 2.82i·8-s + 7.00·9-s − 6.32·12-s + (3 − 2.23i)14-s + 4.00·16-s − 9.89i·18-s + (5.00 + 6.70i)21-s − 1.41i·23-s + 8.94i·24-s + 12.6·27-s + (−3.16 − 4.24i)28-s + ⋯ |
| L(s) = 1 | − 0.999i·2-s + 1.82·3-s − 1.00·4-s − 1.82i·6-s + (0.597 + 0.801i)7-s + 1.00i·8-s + 2.33·9-s − 1.82·12-s + (0.801 − 0.597i)14-s + 1.00·16-s − 2.33i·18-s + (1.09 + 1.46i)21-s − 0.294i·23-s + 1.82i·24-s + 2.43·27-s + (−0.597 − 0.801i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.38388 - 1.19638i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.38388 - 1.19638i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.58 - 2.12i)T \) |
| good | 3 | \( 1 - 3.16T + 3T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 4.47iT - 41T^{2} \) |
| 43 | \( 1 + 12.7iT - 43T^{2} \) |
| 47 | \( 1 + 9.48T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.4iT - 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 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
−10.13430269540600322136672167458, −9.346236067369893767698822864998, −8.719366225392241989235081613819, −8.181859068625568922748851350187, −7.23473190476911365681746903651, −5.50039622003881999209519640717, −4.36350788150771762735323164359, −3.45394285602733715482497904154, −2.47487535979134291505373881433, −1.70921329344568624126499144726,
1.55217289101112456706557976326, 3.19720138033466458782718760430, 4.07735409819160113756172774426, 4.93041736814214159428189312140, 6.47195174453975681492537836677, 7.45841404453499605769812552960, 7.894789578175284498469227294566, 8.610633299165012110879434177761, 9.505118226721568725876823741828, 10.03311840518136367084396978622
