| L(s) = 1 | + 7-s + 4·11-s + 2·13-s + 4·17-s + 4·23-s − 5·25-s − 4·29-s − 8·31-s − 2·37-s − 4·41-s + 8·43-s + 8·47-s + 49-s + 4·53-s + 8·59-s − 2·61-s + 8·67-s + 12·71-s + 6·73-s + 4·77-s − 8·79-s + 16·83-s − 12·89-s + 2·91-s − 2·97-s − 8·103-s + 4·107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s + 0.554·13-s + 0.970·17-s + 0.834·23-s − 25-s − 0.742·29-s − 1.43·31-s − 0.328·37-s − 0.624·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s + 0.549·53-s + 1.04·59-s − 0.256·61-s + 0.977·67-s + 1.42·71-s + 0.702·73-s + 0.455·77-s − 0.900·79-s + 1.75·83-s − 1.27·89-s + 0.209·91-s − 0.203·97-s − 0.788·103-s + 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.086305767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.086305767\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.169196659342650824242531835407, −8.440625678216892840423679471778, −7.53746563106682265921294207979, −6.88806738095765915509810366663, −5.87747399696551577020985934926, −5.27700358495330456685288790654, −4.02277181134495521777926389519, −3.52701351257735111742562589827, −2.07836790041943761710948935981, −1.03480191408612537828984009059,
1.03480191408612537828984009059, 2.07836790041943761710948935981, 3.52701351257735111742562589827, 4.02277181134495521777926389519, 5.27700358495330456685288790654, 5.87747399696551577020985934926, 6.88806738095765915509810366663, 7.53746563106682265921294207979, 8.440625678216892840423679471778, 9.169196659342650824242531835407
