L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s + 12-s − 4·13-s − 2·14-s + 16-s − 6·17-s − 18-s + 2·21-s − 24-s + 4·26-s + 27-s + 2·28-s + 8·29-s − 31-s − 32-s + 6·34-s + 36-s − 4·37-s − 4·39-s + 10·41-s − 2·42-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.436·21-s − 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.377·28-s + 1.48·29-s − 0.179·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.657·37-s − 0.640·39-s + 1.56·41-s − 0.308·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.684305162\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.684305162\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.339728430807411036268799027960, −7.76469596399246854457965500567, −6.98890648667183965663464076727, −6.47522161026533506018183071234, −5.23634413880240840750374326793, −4.63492233429204429534551590778, −3.69921529265913535395438942367, −2.46848149870992704871163585855, −2.11034307770309660356119812362, −0.76748856309929707957793764304,
0.76748856309929707957793764304, 2.11034307770309660356119812362, 2.46848149870992704871163585855, 3.69921529265913535395438942367, 4.63492233429204429534551590778, 5.23634413880240840750374326793, 6.47522161026533506018183071234, 6.98890648667183965663464076727, 7.76469596399246854457965500567, 8.339728430807411036268799027960