L(s) = 1 | + (−0.980 + 0.195i)5-s + (−0.382 − 0.923i)7-s + (0.555 − 0.831i)11-s + (−0.980 − 0.195i)13-s + (−0.707 − 0.707i)17-s + (−0.195 + 0.980i)19-s + (0.923 + 0.382i)23-s + (0.923 − 0.382i)25-s + (−0.555 − 0.831i)29-s − i·31-s + (0.555 + 0.831i)35-s + (0.195 + 0.980i)37-s + (−0.923 − 0.382i)41-s + (0.831 + 0.555i)43-s + (0.707 + 0.707i)47-s + ⋯ |
L(s) = 1 | + (−0.980 + 0.195i)5-s + (−0.382 − 0.923i)7-s + (0.555 − 0.831i)11-s + (−0.980 − 0.195i)13-s + (−0.707 − 0.707i)17-s + (−0.195 + 0.980i)19-s + (0.923 + 0.382i)23-s + (0.923 − 0.382i)25-s + (−0.555 − 0.831i)29-s − i·31-s + (0.555 + 0.831i)35-s + (0.195 + 0.980i)37-s + (−0.923 − 0.382i)41-s + (0.831 + 0.555i)43-s + (0.707 + 0.707i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2903861942 + 0.3366382996i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2903861942 + 0.3366382996i\) |
\(L(1)\) |
\(\approx\) |
\(0.7233216610 - 0.06430149903i\) |
\(L(1)\) |
\(\approx\) |
\(0.7233216610 - 0.06430149903i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.980 + 0.195i)T \) |
| 7 | \( 1 + (-0.382 - 0.923i)T \) |
| 11 | \( 1 + (0.555 - 0.831i)T \) |
| 13 | \( 1 + (-0.980 - 0.195i)T \) |
| 17 | \( 1 + (-0.707 - 0.707i)T \) |
| 19 | \( 1 + (-0.195 + 0.980i)T \) |
| 23 | \( 1 + (0.923 + 0.382i)T \) |
| 29 | \( 1 + (-0.555 - 0.831i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (0.195 + 0.980i)T \) |
| 41 | \( 1 + (-0.923 - 0.382i)T \) |
| 43 | \( 1 + (0.831 + 0.555i)T \) |
| 47 | \( 1 + (0.707 + 0.707i)T \) |
| 53 | \( 1 + (-0.555 + 0.831i)T \) |
| 59 | \( 1 + (0.980 - 0.195i)T \) |
| 61 | \( 1 + (-0.831 + 0.555i)T \) |
| 67 | \( 1 + (-0.831 + 0.555i)T \) |
| 71 | \( 1 + (-0.382 - 0.923i)T \) |
| 73 | \( 1 + (-0.382 + 0.923i)T \) |
| 79 | \( 1 + (-0.707 + 0.707i)T \) |
| 83 | \( 1 + (-0.195 + 0.980i)T \) |
| 89 | \( 1 + (0.923 - 0.382i)T \) |
| 97 | \( 1 - iT \) |
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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
−24.11308364889662919693855523331, −23.18531558292585613354769706979, −22.25789336038987322842096906764, −21.695045401799268547481556093115, −20.34998078791059875979106164782, −19.59480574798818925054242755352, −19.06805560207022263971063534900, −17.88209817641062213435203541229, −16.99365684465151894792617688632, −15.97349073672107200105779894759, −15.11441068648745208809552185570, −14.657244408834454764223998156365, −12.97925169546239391357915037502, −12.39373744045759142885515623281, −11.60480560846156485892054749612, −10.553445145609602593044589240, −9.17725696998074529892881428707, −8.73486396295893546763275979889, −7.31972392260248296011608085052, −6.67152356753093813902048244333, −5.16409224101688911611905007430, −4.35364241838050745673312782931, −3.105948027741878905489438998445, −1.926711523000824407676331889386, −0.15331020078781767469814445584,
0.92905573086665653756760292579, 2.78318572131226737466804562743, 3.784183782732430127681079114, 4.61777605294124079939873432785, 6.10280102165817576184150382010, 7.178201891951699121216899909650, 7.81476906177401837529950168780, 9.03635612781688158798397370374, 10.08628098245124567771458940781, 11.11941372388911928674719165554, 11.79723703733474418589557769674, 12.92083914805313330484237276102, 13.86417172813551070858201696704, 14.79829291655625694081826148007, 15.69203750944781097922999241496, 16.69473903004965606115854416875, 17.21288966620538836336472503163, 18.71330052584409320115789408257, 19.27692231071696426662389540074, 20.07994960156283210556698706543, 20.86551898546172713185352602334, 22.3357571894785264975729090054, 22.61682291697189446564506735051, 23.71982781532015456647107254338, 24.36847864785586571377102552293