| L(s) = 1 | − 5-s − 11-s − 2·13-s + 2·19-s + 9·23-s − 4·25-s + 4·29-s + 5·31-s + 9·37-s + 2·41-s − 6·43-s + 4·47-s − 6·53-s + 55-s − 5·59-s + 2·65-s − 13·67-s − 71-s − 14·73-s + 10·79-s + 14·83-s − 13·89-s − 2·95-s + 19·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.554·13-s + 0.458·19-s + 1.87·23-s − 4/5·25-s + 0.742·29-s + 0.898·31-s + 1.47·37-s + 0.312·41-s − 0.914·43-s + 0.583·47-s − 0.824·53-s + 0.134·55-s − 0.650·59-s + 0.248·65-s − 1.58·67-s − 0.118·71-s − 1.63·73-s + 1.12·79-s + 1.53·83-s − 1.37·89-s − 0.205·95-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.230328097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.230328097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−12.62834876683407, −12.09450621215798, −11.81904140908608, −11.30117858014588, −10.89835259495749, −10.34848991896845, −9.975577811652166, −9.337136038833553, −9.114866936468189, −8.449402227515441, −7.956893832780002, −7.566528511519280, −7.195252789949041, −6.560596780816232, −6.142920301013767, −5.546646748136247, −4.961230085279996, −4.567798897745342, −4.186410287877031, −3.268370373471762, −3.023537362482220, −2.491363144438311, −1.713759766415290, −1.007708041517652, −0.4548360098591615,
0.4548360098591615, 1.007708041517652, 1.713759766415290, 2.491363144438311, 3.023537362482220, 3.268370373471762, 4.186410287877031, 4.567798897745342, 4.961230085279996, 5.546646748136247, 6.142920301013767, 6.560596780816232, 7.195252789949041, 7.566528511519280, 7.956893832780002, 8.449402227515441, 9.114866936468189, 9.337136038833553, 9.975577811652166, 10.34848991896845, 10.89835259495749, 11.30117858014588, 11.81904140908608, 12.09450621215798, 12.62834876683407
