L(s) = 1 | + 4·7-s − 4·11-s + 4·13-s − 6·17-s + 4·19-s + 4·23-s + 4·29-s + 4·37-s − 8·41-s + 12·47-s + 9·49-s − 2·53-s + 12·59-s + 2·61-s − 8·67-s + 8·71-s − 16·73-s − 16·77-s + 8·79-s + 8·83-s + 16·91-s − 8·97-s + 12·101-s + 4·103-s + 8·107-s + 18·109-s − 10·113-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 1.20·11-s + 1.10·13-s − 1.45·17-s + 0.917·19-s + 0.834·23-s + 0.742·29-s + 0.657·37-s − 1.24·41-s + 1.75·47-s + 9/7·49-s − 0.274·53-s + 1.56·59-s + 0.256·61-s − 0.977·67-s + 0.949·71-s − 1.87·73-s − 1.82·77-s + 0.900·79-s + 0.878·83-s + 1.67·91-s − 0.812·97-s + 1.19·101-s + 0.394·103-s + 0.773·107-s + 1.72·109-s − 0.940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.266616162\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.266616162\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.598974870879869589329368450076, −7.85073196695959744960721875463, −7.23308433437246791610041755077, −6.29877617014679624965494671369, −5.34470735103682287937553829653, −4.86559857721531532212898521293, −4.04154767432003311096379164050, −2.88398504463983276447796810886, −1.99605634050645847530772075295, −0.920195558759771845040980682037,
0.920195558759771845040980682037, 1.99605634050645847530772075295, 2.88398504463983276447796810886, 4.04154767432003311096379164050, 4.86559857721531532212898521293, 5.34470735103682287937553829653, 6.29877617014679624965494671369, 7.23308433437246791610041755077, 7.85073196695959744960721875463, 8.598974870879869589329368450076