L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s + 13-s + 2·15-s − 6·17-s + 4·19-s + 21-s − 25-s + 27-s − 2·29-s + 8·31-s + 2·35-s + 6·37-s + 39-s + 6·41-s − 4·43-s + 2·45-s + 12·47-s + 49-s − 6·51-s − 2·53-s + 4·57-s − 2·61-s + 63-s + 2·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.277·13-s + 0.516·15-s − 1.45·17-s + 0.917·19-s + 0.218·21-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.338·35-s + 0.986·37-s + 0.160·39-s + 0.937·41-s − 0.609·43-s + 0.298·45-s + 1.75·47-s + 1/7·49-s − 0.840·51-s − 0.274·53-s + 0.529·57-s − 0.256·61-s + 0.125·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.150203254\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.150203254\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.335026772516990554495888778803, −7.78693608232343540980298539929, −6.86892070531631043279016495423, −6.22250236637411340982406205721, −5.43348359710482436011861456133, −4.58712468210759987003533256725, −3.83115947684622611116307293818, −2.67501609793274540479974054631, −2.11215562610531116890620317025, −1.01699553056073362391268852351,
1.01699553056073362391268852351, 2.11215562610531116890620317025, 2.67501609793274540479974054631, 3.83115947684622611116307293818, 4.58712468210759987003533256725, 5.43348359710482436011861456133, 6.22250236637411340982406205721, 6.86892070531631043279016495423, 7.78693608232343540980298539929, 8.335026772516990554495888778803