L(s) = 1 | + (−0.329 − 0.944i)2-s + (0.0881 + 0.996i)3-s + (−0.783 + 0.621i)4-s + (0.662 + 0.749i)5-s + (0.911 − 0.411i)6-s + (0.713 + 0.700i)7-s + (0.844 + 0.535i)8-s + (−0.984 + 0.175i)9-s + (0.489 − 0.871i)10-s + (−0.949 + 0.312i)11-s + (−0.688 − 0.725i)12-s + (−0.863 + 0.505i)13-s + (0.427 − 0.904i)14-s + (−0.688 + 0.725i)15-s + (0.227 − 0.973i)16-s + (−0.635 − 0.772i)17-s + ⋯ |
L(s) = 1 | + (−0.329 − 0.944i)2-s + (0.0881 + 0.996i)3-s + (−0.783 + 0.621i)4-s + (0.662 + 0.749i)5-s + (0.911 − 0.411i)6-s + (0.713 + 0.700i)7-s + (0.844 + 0.535i)8-s + (−0.984 + 0.175i)9-s + (0.489 − 0.871i)10-s + (−0.949 + 0.312i)11-s + (−0.688 − 0.725i)12-s + (−0.863 + 0.505i)13-s + (0.427 − 0.904i)14-s + (−0.688 + 0.725i)15-s + (0.227 − 0.973i)16-s + (−0.635 − 0.772i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6820427226 + 0.5559826184i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6820427226 + 0.5559826184i\) |
\(L(1)\) |
\(\approx\) |
\(0.8561878962 + 0.2096301033i\) |
\(L(1)\) |
\(\approx\) |
\(0.8561878962 + 0.2096301033i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 179 | \( 1 \) |
good | 2 | \( 1 + (-0.329 - 0.944i)T \) |
| 3 | \( 1 + (0.0881 + 0.996i)T \) |
| 5 | \( 1 + (0.662 + 0.749i)T \) |
| 7 | \( 1 + (0.713 + 0.700i)T \) |
| 11 | \( 1 + (-0.949 + 0.312i)T \) |
| 13 | \( 1 + (-0.863 + 0.505i)T \) |
| 17 | \( 1 + (-0.635 - 0.772i)T \) |
| 19 | \( 1 + (-0.737 - 0.675i)T \) |
| 23 | \( 1 + (0.960 + 0.278i)T \) |
| 29 | \( 1 + (0.295 + 0.955i)T \) |
| 31 | \( 1 + (0.362 + 0.932i)T \) |
| 37 | \( 1 + (0.295 - 0.955i)T \) |
| 41 | \( 1 + (0.990 - 0.140i)T \) |
| 43 | \( 1 + (-0.123 - 0.992i)T \) |
| 47 | \( 1 + (0.880 + 0.474i)T \) |
| 53 | \( 1 + (-0.261 + 0.965i)T \) |
| 59 | \( 1 + (-0.969 + 0.244i)T \) |
| 61 | \( 1 + (0.550 + 0.835i)T \) |
| 67 | \( 1 + (0.844 - 0.535i)T \) |
| 71 | \( 1 + (0.804 + 0.593i)T \) |
| 73 | \( 1 + (0.997 - 0.0705i)T \) |
| 79 | \( 1 + (-0.520 - 0.854i)T \) |
| 83 | \( 1 + (-0.192 - 0.981i)T \) |
| 89 | \( 1 + (-0.329 + 0.944i)T \) |
| 97 | \( 1 + (-0.458 - 0.888i)T \) |
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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
−26.9002530156064966926235202573, −26.03314151883667594820243678525, −24.99069876709569726799109071580, −24.38066618688482781105941167758, −23.73277318892828701275550821603, −22.80843952450390290930493758329, −21.29216720795485427720337457128, −20.19832010333813222598388818745, −19.15628076035078409264085976065, −18.11737642937482221417779770365, −17.20299216503455137622258096067, −16.90432515210227493535538205417, −15.25064697628750422211238536727, −14.25677080681896432432090237577, −13.29666317005551753032577051037, −12.72780674881853854809097215366, −10.9199181597699589212003941664, −9.764544109569965588135040633834, −8.28388773294692139374697914421, −7.96818440292502600185318598069, −6.60455300913404373231978606144, −5.581351312964721231692368989577, −4.54114535872577114897050660396, −2.16618920039578552088798314090, −0.77980912552717900600268243713,
2.27222358992510117081245637644, 2.83151407198072358904571700482, 4.57414596615327127855608392523, 5.32166385028086903569794814608, 7.28812913833154953590109760102, 8.807154156182553642659002900526, 9.45193770588287676426920924904, 10.64662012304194639518676717059, 11.112107716234725486552593922253, 12.41090026946624474580327724190, 13.794525185716172820881393898426, 14.654355878746800088126308713528, 15.6909935399643459381567452501, 17.2018184017115750661636697502, 17.84646916921213771412267262063, 18.837275354520289317207530433114, 19.9674537919109686928090174227, 21.13230748199957629357940463341, 21.51206976845770361519112041828, 22.27364787978625631256276517966, 23.34514519430896652770241321083, 25.07465112685186215797862032194, 26.00463217419677917017035391164, 26.797938985688124425435841878264, 27.47512125537165287242597494987