L(s) = 1 | + 4·11-s − 6·13-s − 6·17-s + 4·19-s − 4·23-s + 2·29-s − 8·31-s − 6·37-s + 6·41-s − 8·43-s + 6·53-s − 4·59-s − 10·61-s − 8·67-s − 12·71-s − 14·73-s − 16·79-s + 12·83-s + 14·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 1.20·11-s − 1.66·13-s − 1.45·17-s + 0.917·19-s − 0.834·23-s + 0.371·29-s − 1.43·31-s − 0.986·37-s + 0.937·41-s − 1.21·43-s + 0.824·53-s − 0.520·59-s − 1.28·61-s − 0.977·67-s − 1.42·71-s − 1.63·73-s − 1.80·79-s + 1.31·83-s + 1.48·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2633752327\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2633752327\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.45173512847189, −11.96363227670529, −11.75567961256999, −11.34876055979332, −10.66909423517734, −10.19323319723680, −9.912191928113178, −9.155631023173345, −9.021109025099651, −8.665323951828753, −7.768687545079904, −7.302744744479118, −7.239849926382561, −6.462819971268517, −6.109151141338694, −5.547273877735207, −4.799885616575777, −4.655656330862911, −4.010202324404742, −3.463104013360781, −2.903323500438720, −2.226907161250376, −1.798088761090895, −1.190448436843432, −0.1308642987711358,
0.1308642987711358, 1.190448436843432, 1.798088761090895, 2.226907161250376, 2.903323500438720, 3.463104013360781, 4.010202324404742, 4.655656330862911, 4.799885616575777, 5.547273877735207, 6.109151141338694, 6.462819971268517, 7.239849926382561, 7.302744744479118, 7.768687545079904, 8.665323951828753, 9.021109025099651, 9.155631023173345, 9.912191928113178, 10.19323319723680, 10.66909423517734, 11.34876055979332, 11.75567961256999, 11.96363227670529, 12.45173512847189