L(s) = 1 | + 5-s + 7-s + 13-s + 17-s − 2·31-s + 35-s − 37-s + 43-s − 47-s + 65-s − 71-s + 85-s + 91-s + 2·107-s − 109-s − 2·113-s + 119-s + ⋯ |
L(s) = 1 | + 5-s + 7-s + 13-s + 17-s − 2·31-s + 35-s − 37-s + 43-s − 47-s + 65-s − 71-s + 85-s + 91-s + 2·107-s − 109-s − 2·113-s + 119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.832426030\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.832426030\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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
−8.743482093758118961320489667409, −7.967386679435940305650168263944, −7.30898559345027008254216268681, −6.33190715670784493096816996215, −5.61260268309217016152046500407, −5.16702174510769087330189145955, −4.07034202281980020039752522804, −3.21753954892597903232864497630, −1.96969571707703874645104744886, −1.38198762194799550133640712557,
1.38198762194799550133640712557, 1.96969571707703874645104744886, 3.21753954892597903232864497630, 4.07034202281980020039752522804, 5.16702174510769087330189145955, 5.61260268309217016152046500407, 6.33190715670784493096816996215, 7.30898559345027008254216268681, 7.967386679435940305650168263944, 8.743482093758118961320489667409