L(s) = 1 | − 3-s + 9-s − 2·11-s − 5·13-s − 2·17-s − 3·19-s + 2·23-s − 5·25-s − 27-s − 8·29-s − 31-s + 2·33-s + 5·37-s + 5·39-s + 2·41-s − 7·43-s − 8·47-s + 2·51-s + 2·53-s + 3·57-s − 10·59-s + 2·61-s + 11·67-s − 2·69-s + 12·71-s − 3·73-s + 5·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.38·13-s − 0.485·17-s − 0.688·19-s + 0.417·23-s − 25-s − 0.192·27-s − 1.48·29-s − 0.179·31-s + 0.348·33-s + 0.821·37-s + 0.800·39-s + 0.312·41-s − 1.06·43-s − 1.16·47-s + 0.280·51-s + 0.274·53-s + 0.397·57-s − 1.30·59-s + 0.256·61-s + 1.34·67-s − 0.240·69-s + 1.42·71-s − 0.351·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6673869601\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6673869601\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.83141925498596287794233179508, −6.90250593677208284838758087058, −6.44057780658717800521746521491, −5.49723376123085676872274712540, −5.06403904214420508858292550744, −4.33492905257565514745921200520, −3.51388843962106887870279566118, −2.44391351638937028875151774657, −1.84950194412471397163540940870, −0.38195678575605166756051197737,
0.38195678575605166756051197737, 1.84950194412471397163540940870, 2.44391351638937028875151774657, 3.51388843962106887870279566118, 4.33492905257565514745921200520, 5.06403904214420508858292550744, 5.49723376123085676872274712540, 6.44057780658717800521746521491, 6.90250593677208284838758087058, 7.83141925498596287794233179508