L(s) = 1 | + (0.120 − 0.992i)2-s + (0.354 + 0.935i)3-s + (−0.970 − 0.239i)4-s + (0.885 − 0.464i)5-s + (0.970 − 0.239i)6-s + (0.354 + 0.935i)7-s + (−0.354 + 0.935i)8-s + (−0.748 + 0.663i)9-s + (−0.354 − 0.935i)10-s + (0.885 + 0.464i)11-s + (−0.120 − 0.992i)12-s + (−0.970 − 0.239i)13-s + (0.970 − 0.239i)14-s + (0.748 + 0.663i)15-s + (0.885 + 0.464i)16-s + (0.970 + 0.239i)17-s + ⋯ |
L(s) = 1 | + (0.120 − 0.992i)2-s + (0.354 + 0.935i)3-s + (−0.970 − 0.239i)4-s + (0.885 − 0.464i)5-s + (0.970 − 0.239i)6-s + (0.354 + 0.935i)7-s + (−0.354 + 0.935i)8-s + (−0.748 + 0.663i)9-s + (−0.354 − 0.935i)10-s + (0.885 + 0.464i)11-s + (−0.120 − 0.992i)12-s + (−0.970 − 0.239i)13-s + (0.970 − 0.239i)14-s + (0.748 + 0.663i)15-s + (0.885 + 0.464i)16-s + (0.970 + 0.239i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.079946150 + 0.1058330950i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.079946150 + 0.1058330950i\) |
\(L(1)\) |
\(\approx\) |
\(1.400108014 - 0.1173026790i\) |
\(L(1)\) |
\(\approx\) |
\(1.400108014 - 0.1173026790i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 79 | \( 1 \) |
good | 2 | \( 1 + (0.120 - 0.992i)T \) |
| 3 | \( 1 + (0.354 + 0.935i)T \) |
| 5 | \( 1 + (0.885 - 0.464i)T \) |
| 7 | \( 1 + (0.354 + 0.935i)T \) |
| 11 | \( 1 + (0.885 + 0.464i)T \) |
| 13 | \( 1 + (-0.970 - 0.239i)T \) |
| 17 | \( 1 + (0.970 + 0.239i)T \) |
| 19 | \( 1 + (0.568 + 0.822i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.748 + 0.663i)T \) |
| 31 | \( 1 + (0.120 - 0.992i)T \) |
| 37 | \( 1 + (-0.568 - 0.822i)T \) |
| 41 | \( 1 + (-0.885 + 0.464i)T \) |
| 43 | \( 1 + (-0.885 + 0.464i)T \) |
| 47 | \( 1 + (-0.568 + 0.822i)T \) |
| 53 | \( 1 + (0.354 - 0.935i)T \) |
| 59 | \( 1 + (0.970 - 0.239i)T \) |
| 61 | \( 1 + (-0.568 - 0.822i)T \) |
| 67 | \( 1 + (0.120 + 0.992i)T \) |
| 71 | \( 1 + (0.354 - 0.935i)T \) |
| 73 | \( 1 + (-0.970 + 0.239i)T \) |
| 83 | \( 1 + (-0.970 - 0.239i)T \) |
| 89 | \( 1 + (-0.354 - 0.935i)T \) |
| 97 | \( 1 + (0.568 + 0.822i)T \) |
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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
−30.722561752589645195087400931712, −30.036579251538608846451709572646, −29.01530356990108260316549418491, −27.140305928858171154251094902550, −26.37541976695611423686919484587, −25.24781289243443264487857276450, −24.58696635270105904932961425820, −23.55496415765209684005509838262, −22.49657434261530626349680661171, −21.267059449985198584386986081535, −19.643060673105710854228841046858, −18.54688789346450627410625320356, −17.3777646737766204692172306652, −16.90973049687433369892205343540, −14.85660914331626707878886105038, −14.02227833678990806945694440840, −13.470018315807734794647049609387, −11.91051702146642858502126439699, −9.968307362046017103376985278049, −8.71803645456520929444943965876, −7.24672252533957715087313401956, −6.67716287746779968295884246595, −5.158650441189307244866219028441, −3.20343577177663372853064418164, −1.08615395631841161439111259890,
1.72837899927658469698956429773, 3.0858696643192084638248840189, 4.74181880226451311626491552834, 5.58371825053611653845529144763, 8.39562844241219515772241608974, 9.46734266033778409101367809817, 10.07329167597503480427953618290, 11.667672963493604680178285773381, 12.68451142734974393478439912888, 14.26043956361751811138520344257, 14.84143672855190877234677659823, 16.71210245984114065457863688494, 17.70244476290703643504682560344, 19.11936774154277788768360211476, 20.278818456407404943473785478472, 21.15133006927414015975526550762, 21.86561565893192986953839025575, 22.74634342722237550757444019747, 24.663841431135428355219592857032, 25.53347686761274989465654862022, 27.04379971931495359807239584602, 27.79429740574727733444471120317, 28.637896010706891349631753890506, 29.736030908497031659284100846012, 31.02495579096886406423752681885