L(s) = 1 | + 3·3-s + 3·5-s + 9·9-s + 15·11-s + 64·13-s + 9·15-s − 84·17-s − 16·19-s + 84·23-s − 116·25-s + 27·27-s − 297·29-s − 253·31-s + 45·33-s − 316·37-s + 192·39-s − 360·41-s − 26·43-s + 27·45-s − 30·47-s − 252·51-s + 363·53-s + 45·55-s − 48·57-s − 15·59-s + 118·61-s + 192·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.268·5-s + 1/3·9-s + 0.411·11-s + 1.36·13-s + 0.154·15-s − 1.19·17-s − 0.193·19-s + 0.761·23-s − 0.927·25-s + 0.192·27-s − 1.90·29-s − 1.46·31-s + 0.237·33-s − 1.40·37-s + 0.788·39-s − 1.37·41-s − 0.0922·43-s + 0.0894·45-s − 0.0931·47-s − 0.691·51-s + 0.940·53-s + 0.110·55-s − 0.111·57-s − 0.0330·59-s + 0.247·61-s + 0.366·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 15 T + p^{3} T^{2} \) |
| 13 | \( 1 - 64 T + p^{3} T^{2} \) |
| 17 | \( 1 + 84 T + p^{3} T^{2} \) |
| 19 | \( 1 + 16 T + p^{3} T^{2} \) |
| 23 | \( 1 - 84 T + p^{3} T^{2} \) |
| 29 | \( 1 + 297 T + p^{3} T^{2} \) |
| 31 | \( 1 + 253 T + p^{3} T^{2} \) |
| 37 | \( 1 + 316 T + p^{3} T^{2} \) |
| 41 | \( 1 + 360 T + p^{3} T^{2} \) |
| 43 | \( 1 + 26 T + p^{3} T^{2} \) |
| 47 | \( 1 + 30 T + p^{3} T^{2} \) |
| 53 | \( 1 - 363 T + p^{3} T^{2} \) |
| 59 | \( 1 + 15 T + p^{3} T^{2} \) |
| 61 | \( 1 - 118 T + p^{3} T^{2} \) |
| 67 | \( 1 - 370 T + p^{3} T^{2} \) |
| 71 | \( 1 - 342 T + p^{3} T^{2} \) |
| 73 | \( 1 + 362 T + p^{3} T^{2} \) |
| 79 | \( 1 + 467 T + p^{3} T^{2} \) |
| 83 | \( 1 - 477 T + p^{3} T^{2} \) |
| 89 | \( 1 + 906 T + p^{3} T^{2} \) |
| 97 | \( 1 + 503 T + p^{3} T^{2} \) |
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
−8.538106442656313879908579130193, −7.41945651372202687786155297726, −6.79221194929837980216803206923, −5.93777594421574156551358414107, −5.11358253552231939908642827230, −3.88716262973312236521405538879, −3.52940137605478574098691127308, −2.14220364139252212553734794780, −1.50692040480369034830262916231, 0,
1.50692040480369034830262916231, 2.14220364139252212553734794780, 3.52940137605478574098691127308, 3.88716262973312236521405538879, 5.11358253552231939908642827230, 5.93777594421574156551358414107, 6.79221194929837980216803206923, 7.41945651372202687786155297726, 8.538106442656313879908579130193