L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.415 + 0.909i)3-s + (−0.142 + 0.989i)4-s + (0.959 + 0.281i)5-s + (−0.959 + 0.281i)6-s + (0.654 + 0.755i)7-s + (−0.841 + 0.540i)8-s + (−0.654 − 0.755i)9-s + (0.415 + 0.909i)10-s + (0.959 + 0.281i)11-s + (−0.841 − 0.540i)12-s + (−0.841 − 0.540i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (−0.959 − 0.281i)16-s + (−0.142 − 0.989i)17-s + ⋯ |
L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.415 + 0.909i)3-s + (−0.142 + 0.989i)4-s + (0.959 + 0.281i)5-s + (−0.959 + 0.281i)6-s + (0.654 + 0.755i)7-s + (−0.841 + 0.540i)8-s + (−0.654 − 0.755i)9-s + (0.415 + 0.909i)10-s + (0.959 + 0.281i)11-s + (−0.841 − 0.540i)12-s + (−0.841 − 0.540i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (−0.959 − 0.281i)16-s + (−0.142 − 0.989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4892271064 + 2.173754521i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4892271064 + 2.173754521i\) |
\(L(1)\) |
\(\approx\) |
\(0.9757341412 + 1.191489898i\) |
\(L(1)\) |
\(\approx\) |
\(0.9757341412 + 1.191489898i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 67 | \( 1 \) |
good | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 5 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (0.654 + 0.755i)T \) |
| 11 | \( 1 + (0.959 + 0.281i)T \) |
| 13 | \( 1 + (-0.841 - 0.540i)T \) |
| 17 | \( 1 + (-0.142 - 0.989i)T \) |
| 19 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (0.415 - 0.909i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (-0.841 + 0.540i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.142 + 0.989i)T \) |
| 43 | \( 1 + (0.142 + 0.989i)T \) |
| 47 | \( 1 + (0.415 - 0.909i)T \) |
| 53 | \( 1 + (0.142 - 0.989i)T \) |
| 59 | \( 1 + (0.841 - 0.540i)T \) |
| 61 | \( 1 + (0.959 - 0.281i)T \) |
| 71 | \( 1 + (-0.142 + 0.989i)T \) |
| 73 | \( 1 + (-0.959 + 0.281i)T \) |
| 79 | \( 1 + (-0.841 - 0.540i)T \) |
| 83 | \( 1 + (-0.959 - 0.281i)T \) |
| 89 | \( 1 + (0.415 + 0.909i)T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−30.90693774767611239963871639358, −30.01873129058858299304832428037, −29.40210005497917157618369530989, −28.43433043914894156473135835641, −27.26185565678658409738021626823, −25.40688950609650075704326061663, −24.19124211470299774690695448416, −23.72114550914326186726364866021, −22.19032481153384322152993894572, −21.43746345365943805144021079648, −19.998101603211749710777698984276, −19.155873725784678403634622596777, −17.624812235962011309992859456779, −16.99527349309525461751425123000, −14.59336434716889384256663567575, −13.76766033824134628817337366251, −12.81541568219909224446924026507, −11.60613914662326662208536106204, −10.53943416312599974236421288582, −8.98904502678760525331534245803, −6.94840690155216206500947661524, −5.76086065307963205921268056392, −4.40308678640736028450713040806, −2.17652015839765108608257371865, −1.07754618106825380392565727177,
2.7149539546885909365695552442, 4.56574090835911586405568039972, 5.52955636942989397525429881015, 6.67928947542098951361700975290, 8.63333756333185562583096619800, 9.80287428529523709942898426854, 11.44569108686949086942190036336, 12.609745905253911236559683057399, 14.53313973286548084027809144141, 14.706940310326571044983994308453, 16.28230077707637428035454005393, 17.312157375117300097259094340533, 18.10173452079142720791796737035, 20.46965551252696208886034476278, 21.53309880415113216079202391308, 22.14523131608543765898101918656, 23.08764295044750956457881751767, 24.823258435330751086627508476696, 25.246395858787299032796248783847, 26.76742853816583338372525706735, 27.52047733492426598300972973480, 29.02950454389454958295809676744, 30.1483285492744245378590602497, 31.462036470407930478012226070606, 32.45716603563462343050579485923