L(s) = 1 | + (0.517 + 0.517i)5-s + 1.41·7-s + (2.73 − 2.73i)11-s + (−1.73 − 1.73i)13-s − 0.378i·17-s + (−0.378 + 0.378i)19-s + 3.46i·23-s − 4.46i·25-s + (4.76 − 4.76i)29-s + 0.656i·31-s + (0.732 + 0.732i)35-s + (4.46 − 4.46i)37-s + 10.1·41-s + (−6.31 − 6.31i)43-s − 10.3·47-s + ⋯ |
L(s) = 1 | + (0.231 + 0.231i)5-s + 0.534·7-s + (0.823 − 0.823i)11-s + (−0.480 − 0.480i)13-s − 0.0919i·17-s + (−0.0869 + 0.0869i)19-s + 0.722i·23-s − 0.892i·25-s + (0.883 − 0.883i)29-s + 0.117i·31-s + (0.123 + 0.123i)35-s + (0.733 − 0.733i)37-s + 1.58·41-s + (−0.962 − 0.962i)43-s − 1.51·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.979863644\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.979863644\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.517 - 0.517i)T + 5iT^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + (-2.73 + 2.73i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.73 + 1.73i)T + 13iT^{2} \) |
| 17 | \( 1 + 0.378iT - 17T^{2} \) |
| 19 | \( 1 + (0.378 - 0.378i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 + (-4.76 + 4.76i)T - 29iT^{2} \) |
| 31 | \( 1 - 0.656iT - 31T^{2} \) |
| 37 | \( 1 + (-4.46 + 4.46i)T - 37iT^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + (6.31 + 6.31i)T + 43iT^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + (-4.00 - 4.00i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.92 + 4.92i)T - 59iT^{2} \) |
| 61 | \( 1 + (3 + 3i)T + 61iT^{2} \) |
| 67 | \( 1 + (1.03 - 1.03i)T - 67iT^{2} \) |
| 71 | \( 1 - 14iT - 71T^{2} \) |
| 73 | \( 1 + 8.92iT - 73T^{2} \) |
| 79 | \( 1 - 11.9iT - 79T^{2} \) |
| 83 | \( 1 + (7.26 + 7.26i)T + 83iT^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 2.39T + 97T^{2} \) |
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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
−8.844263618791752874789038025410, −8.163219375521942426799104684806, −7.45865039470377587119357573440, −6.46336056034256321505378596162, −5.88300988990427347826415116560, −4.93960569245121203959550168451, −4.05184130594857893667808780423, −3.07439284517616794696010746991, −2.05601798227476941453800155263, −0.75693794620262642423602221787,
1.24061885909313295139051244733, 2.13883241054872638421001066532, 3.33539397811765065158188884433, 4.61043871273210324845486766250, 4.78964635867643245606840610874, 6.08331200557532388275875768234, 6.76782640390712901106810439691, 7.53572349281249740860502380739, 8.384672440753406224782202478584, 9.146926279888853984919003615825