| L(s) = 1 | − 2·5-s + 7-s + 4·11-s + 6·13-s + 2·17-s + 4·19-s − 4·23-s − 25-s − 2·29-s − 8·31-s − 2·35-s + 10·37-s + 2·41-s + 8·43-s + 49-s − 10·53-s − 8·55-s + 12·59-s − 10·61-s − 12·65-s − 8·67-s + 12·71-s + 2·73-s + 4·77-s − 12·83-s − 4·85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 1.20·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.338·35-s + 1.64·37-s + 0.312·41-s + 1.21·43-s + 1/7·49-s − 1.37·53-s − 1.07·55-s + 1.56·59-s − 1.28·61-s − 1.48·65-s − 0.977·67-s + 1.42·71-s + 0.234·73-s + 0.455·77-s − 1.31·83-s − 0.433·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.014469483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.014469483\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.320383312015219226294100843388, −7.77853019762625190716075044614, −7.12476921432385806506284419378, −6.09664756569615751308461657816, −5.68030407373648152373224072684, −4.38156718899132832416834458068, −3.87697779819423025511035869420, −3.24828556873724268658050054220, −1.76295140196623459214733320395, −0.871945477605480385408643863818,
0.871945477605480385408643863818, 1.76295140196623459214733320395, 3.24828556873724268658050054220, 3.87697779819423025511035869420, 4.38156718899132832416834458068, 5.68030407373648152373224072684, 6.09664756569615751308461657816, 7.12476921432385806506284419378, 7.77853019762625190716075044614, 8.320383312015219226294100843388
