L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 2·11-s + 4·13-s − 15-s + 2·17-s + 2·19-s − 21-s − 4·23-s + 25-s − 27-s − 2·29-s + 6·31-s + 2·33-s + 35-s − 6·37-s − 4·39-s + 6·41-s + 4·43-s + 45-s + 49-s − 2·51-s + 8·53-s − 2·55-s − 2·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s − 0.258·15-s + 0.485·17-s + 0.458·19-s − 0.218·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.07·31-s + 0.348·33-s + 0.169·35-s − 0.986·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.280·51-s + 1.09·53-s − 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.654840586\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.654840586\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−9.468585753424041859804152269418, −8.481553612925985512669604062490, −7.81835639585916182571375219056, −6.87259354452381471297775882823, −5.94033866886860292935603183313, −5.45803650924575687262352989502, −4.46195444778293602014826196668, −3.43992207551315519554345327677, −2.15715671918253819106635126062, −0.959800325238339118056269866099,
0.959800325238339118056269866099, 2.15715671918253819106635126062, 3.43992207551315519554345327677, 4.46195444778293602014826196668, 5.45803650924575687262352989502, 5.94033866886860292935603183313, 6.87259354452381471297775882823, 7.81835639585916182571375219056, 8.481553612925985512669604062490, 9.468585753424041859804152269418