L(s) = 1 | + (−0.0950 + 0.995i)2-s + (0.981 − 0.189i)3-s + (−0.981 − 0.189i)4-s + (−0.281 + 0.959i)5-s + (0.0950 + 0.995i)6-s + (−0.971 + 0.235i)7-s + (0.281 − 0.959i)8-s + (0.928 − 0.371i)9-s + (−0.928 − 0.371i)10-s + (0.971 + 0.235i)11-s − 12-s + (−0.142 − 0.989i)14-s + (−0.0950 + 0.995i)15-s + (0.928 + 0.371i)16-s + (0.995 − 0.0950i)17-s + (0.281 + 0.959i)18-s + ⋯ |
L(s) = 1 | + (−0.0950 + 0.995i)2-s + (0.981 − 0.189i)3-s + (−0.981 − 0.189i)4-s + (−0.281 + 0.959i)5-s + (0.0950 + 0.995i)6-s + (−0.971 + 0.235i)7-s + (0.281 − 0.959i)8-s + (0.928 − 0.371i)9-s + (−0.928 − 0.371i)10-s + (0.971 + 0.235i)11-s − 12-s + (−0.142 − 0.989i)14-s + (−0.0950 + 0.995i)15-s + (0.928 + 0.371i)16-s + (0.995 − 0.0950i)17-s + (0.281 + 0.959i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.786872710 + 0.005861862962i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.786872710 + 0.005861862962i\) |
\(L(1)\) |
\(\approx\) |
\(1.037535814 + 0.4805350754i\) |
\(L(1)\) |
\(\approx\) |
\(1.037535814 + 0.4805350754i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 89 | \( 1 \) |
good | 2 | \( 1 + (-0.0950 + 0.995i)T \) |
| 3 | \( 1 + (0.981 - 0.189i)T \) |
| 5 | \( 1 + (-0.281 + 0.959i)T \) |
| 7 | \( 1 + (-0.971 + 0.235i)T \) |
| 11 | \( 1 + (0.971 + 0.235i)T \) |
| 17 | \( 1 + (0.995 - 0.0950i)T \) |
| 19 | \( 1 + (-0.371 - 0.928i)T \) |
| 23 | \( 1 + (0.786 + 0.618i)T \) |
| 29 | \( 1 + (0.723 - 0.690i)T \) |
| 31 | \( 1 + (-0.989 + 0.142i)T \) |
| 37 | \( 1 + (-0.866 - 0.5i)T \) |
| 41 | \( 1 + (-0.945 + 0.327i)T \) |
| 43 | \( 1 + (-0.723 - 0.690i)T \) |
| 47 | \( 1 + (-0.755 - 0.654i)T \) |
| 53 | \( 1 + (-0.654 - 0.755i)T \) |
| 59 | \( 1 + (-0.189 + 0.981i)T \) |
| 61 | \( 1 + (-0.888 - 0.458i)T \) |
| 67 | \( 1 + (-0.189 - 0.981i)T \) |
| 71 | \( 1 + (0.690 - 0.723i)T \) |
| 73 | \( 1 + (0.989 - 0.142i)T \) |
| 79 | \( 1 + (-0.142 - 0.989i)T \) |
| 83 | \( 1 + (-0.909 - 0.415i)T \) |
| 97 | \( 1 + (0.971 - 0.235i)T \) |
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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
−20.91723514672073297707792700177, −20.20220849090243118669830286961, −19.70714989171890731094466248643, −19.05385097773985502252218983970, −18.502880598659782442212223139263, −16.9165614678121616996970649676, −16.73195976980080771526968192886, −15.7031558883543771132831603292, −14.53215803029528968031532383298, −14.03216402704575379995955789796, −12.98507547696690758718326582223, −12.61876522490684120483262092045, −11.85591661788347810891359646167, −10.62442480787279982774728967231, −9.86797306705231785131889628147, −9.226338380399297878882801365272, −8.58511563486121122750852073080, −7.87133686068146110571816097590, −6.669854822922649308902609785318, −5.31261954358303290329767215654, −4.34395517567285154919993361522, −3.58359065642871022146035912152, −3.08388804819053913842751239139, −1.667644458658333376146087511099, −1.01803457036268689213822703239,
0.36726616381445653410363335347, 1.81969726427103811462644224574, 3.29025188427928650525454846697, 3.50130164425367057963111610527, 4.74401531957591701630849339420, 6.05176010356593292195452077388, 6.87980674920470738498330130464, 7.17270133213204879550451850812, 8.22111754232581534073675577017, 9.09633960225507653737812888227, 9.6701639755583623901609250116, 10.44644629397028289427644645261, 11.85736874367278868673235017838, 12.73140346348358338286104309165, 13.61734538410137461341434864245, 14.1688367954907981846774639815, 15.154610500349423572228358482544, 15.255831104969341747027789610669, 16.29756884033643833417413644966, 17.168721131704771454606601009240, 18.13676028457963305923700311040, 18.80633258345244667164965503210, 19.44670302774242917210541526455, 19.8246169845075626716351723881, 21.36482729730331412066735394867