| L(s) = 1 | + 3.24i·5-s − 1.54i·7-s + 1.41·11-s + 13-s + 6.43i·17-s − 4.13i·19-s + 4.29·23-s − 5.51·25-s + 1.41i·29-s + 3.54i·31-s + 5.01·35-s − 4.51·37-s + 0.413i·41-s + 6.51i·43-s + 7.89·47-s + ⋯ |
| L(s) = 1 | + 1.44i·5-s − 0.584i·7-s + 0.426·11-s + 0.277·13-s + 1.55i·17-s − 0.948i·19-s + 0.895·23-s − 1.10·25-s + 0.262i·29-s + 0.637i·31-s + 0.848·35-s − 0.741·37-s + 0.0645i·41-s + 0.992i·43-s + 1.15·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.744209561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.744209561\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.24iT - 5T^{2} \) |
| 7 | \( 1 + 1.54iT - 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 17 | \( 1 - 6.43iT - 17T^{2} \) |
| 19 | \( 1 + 4.13iT - 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 3.54iT - 31T^{2} \) |
| 37 | \( 1 + 4.51T + 37T^{2} \) |
| 41 | \( 1 - 0.413iT - 41T^{2} \) |
| 43 | \( 1 - 6.51iT - 43T^{2} \) |
| 47 | \( 1 - 7.89T + 47T^{2} \) |
| 53 | \( 1 + 0.774iT - 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 + 6.13iT - 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 - 1.67T + 73T^{2} \) |
| 79 | \( 1 - 2.07iT - 79T^{2} \) |
| 83 | \( 1 + 4.88T + 83T^{2} \) |
| 89 | \( 1 - 8.89iT - 89T^{2} \) |
| 97 | \( 1 + 9.67T + 97T^{2} \) |
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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
−8.710625204141593896333052804367, −7.86093633381016743122346543819, −7.06401193969944347558595380967, −6.64751338750849118097865622551, −5.98262762027321186460227143390, −4.89053153557688417564353933613, −3.89108530691523301916082094364, −3.31948757308841191242854533051, −2.40116188421931699371346115124, −1.21391662036573359178607394811,
0.56344673357917619559140846370, 1.55435909735985120769569392175, 2.65034795114589978293826482361, 3.77712554071257107722766057395, 4.59240081013944203067856193376, 5.33298122938863751249985040098, 5.81684441724807529430693335259, 6.89377353772007738763355433586, 7.66167288910704567643876816064, 8.508287711904597060625590157696
