L(s) = 1 | − 3-s + 5-s + 9-s + 13-s − 15-s + 2·17-s + 4·23-s + 25-s − 27-s − 6·29-s + 8·31-s − 10·37-s − 39-s − 6·41-s − 4·43-s + 45-s − 8·47-s − 7·49-s − 2·51-s − 2·53-s − 6·61-s + 65-s − 12·67-s − 4·69-s + 8·71-s + 6·73-s − 75-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 0.485·17-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 1.64·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s − 49-s − 0.280·51-s − 0.274·53-s − 0.768·61-s + 0.124·65-s − 1.46·67-s − 0.481·69-s + 0.949·71-s + 0.702·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.72703744639065, −16.16652884049546, −15.35518303965535, −15.10144190152596, −14.29300393641957, −13.65551701100799, −13.28152938324184, −12.56212330624923, −12.05658556026194, −11.44546600120969, −10.85654851515303, −10.30435331447386, −9.715501622567799, −9.162531466435019, −8.371654595122781, −7.833356249553535, −6.888398243766097, −6.569955102215190, −5.785067755082558, −5.138947790489691, −4.677151344697214, −3.607010159437114, −3.045208817578382, −1.899537738679919, −1.226547260975463, 0,
1.226547260975463, 1.899537738679919, 3.045208817578382, 3.607010159437114, 4.677151344697214, 5.138947790489691, 5.785067755082558, 6.569955102215190, 6.888398243766097, 7.833356249553535, 8.371654595122781, 9.162531466435019, 9.715501622567799, 10.30435331447386, 10.85654851515303, 11.44546600120969, 12.05658556026194, 12.56212330624923, 13.28152938324184, 13.65551701100799, 14.29300393641957, 15.10144190152596, 15.35518303965535, 16.16652884049546, 16.72703744639065