L(s) = 1 | − 7-s + (1.5 + 0.866i)19-s + 25-s + (1.5 + 0.866i)31-s + (1 − 1.73i)37-s + (−0.5 + 0.866i)43-s + 49-s + (1.5 − 0.866i)61-s + (−1 + 1.73i)67-s + (−1.5 + 0.866i)73-s + (−1 − 1.73i)79-s + (1.5 + 0.866i)97-s + (−0.5 − 0.866i)109-s + ⋯ |
L(s) = 1 | − 7-s + (1.5 + 0.866i)19-s + 25-s + (1.5 + 0.866i)31-s + (1 − 1.73i)37-s + (−0.5 + 0.866i)43-s + 49-s + (1.5 − 0.866i)61-s + (−1 + 1.73i)67-s + (−1.5 + 0.866i)73-s + (−1 − 1.73i)79-s + (1.5 + 0.866i)97-s + (−0.5 − 0.866i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.101905686\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.101905686\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.294327477107004576495528176382, −8.537055831210476786177897564355, −7.61641572034219727366089564199, −6.93114040587752945451741525712, −6.11255882673049950569124607722, −5.39734720768433834784366247044, −4.36095524852641853657839353913, −3.36974865705466415476703235333, −2.68060789622532968871999362027, −1.12886443883065970027101895307,
0.980121264161258131646835879584, 2.65996633537288379440909000185, 3.22835535128256897174525249439, 4.38289635081508370721195329329, 5.21457309656645287248648761666, 6.17625995762833166141827655367, 6.81422556148650222328618502872, 7.56422410661195451642873117107, 8.486742116533799096673114874733, 9.261796085787225578427612449624