| L(s) = 1 | + i·2-s + 3-s − 4-s + i·5-s + i·6-s − 0.554i·7-s − i·8-s + 9-s − 10-s − 1.35i·11-s − 12-s + 0.554·14-s + i·15-s + 16-s − 1.80·17-s + i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s − 0.209i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.409i·11-s − 0.288·12-s + 0.148·14-s + 0.258i·15-s + 0.250·16-s − 0.437·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.091145763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.091145763\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 0.554iT - 7T^{2} \) |
| 11 | \( 1 + 1.35iT - 11T^{2} \) |
| 17 | \( 1 + 1.80T + 17T^{2} \) |
| 19 | \( 1 - 3.33iT - 19T^{2} \) |
| 23 | \( 1 - 0.692T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 - 1.75iT - 31T^{2} \) |
| 37 | \( 1 - 0.158iT - 37T^{2} \) |
| 41 | \( 1 + 0.664iT - 41T^{2} \) |
| 43 | \( 1 - 3.96T + 43T^{2} \) |
| 47 | \( 1 - 2.10iT - 47T^{2} \) |
| 53 | \( 1 - 1.15T + 53T^{2} \) |
| 59 | \( 1 - 8.72iT - 59T^{2} \) |
| 61 | \( 1 + 5.56T + 61T^{2} \) |
| 67 | \( 1 - 3.58iT - 67T^{2} \) |
| 71 | \( 1 - 4.96iT - 71T^{2} \) |
| 73 | \( 1 + 3.70iT - 73T^{2} \) |
| 79 | \( 1 - 3.25T + 79T^{2} \) |
| 83 | \( 1 - 3.85iT - 83T^{2} \) |
| 89 | \( 1 + 3.13iT - 89T^{2} \) |
| 97 | \( 1 - 15.8iT - 97T^{2} \) |
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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
−8.414083408822735521164872923906, −7.64641099282822668796391450195, −7.11213073718840588534693850170, −6.34058982579208439739512675214, −5.72597618093723185797332935664, −4.73273336889845797358639179730, −3.98943821474265216107456935233, −3.21897176794877205058099768335, −2.31496352406142050048716002246, −1.02400202235327133038177864624,
0.60062776880591691273033623984, 1.77914692444168124076394746784, 2.53308543290099184876610628448, 3.33965599897169529337572240737, 4.33204386115537493807238698313, 4.77807185940953645619772041644, 5.71443775218101947034429044461, 6.66894717292711246109260542628, 7.44382466249461864874060765451, 8.242859903588009542111620638405
