L(s) = 1 | − 3-s + 2·7-s + 9-s − 2·11-s − 13-s − 6·17-s + 2·19-s − 2·21-s − 27-s − 2·29-s + 2·31-s + 2·33-s + 10·37-s + 39-s + 2·41-s + 8·43-s + 2·47-s − 3·49-s + 6·51-s − 2·53-s − 2·57-s − 10·59-s − 6·61-s + 2·63-s + 14·67-s − 14·71-s + 6·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 1.45·17-s + 0.458·19-s − 0.436·21-s − 0.192·27-s − 0.371·29-s + 0.359·31-s + 0.348·33-s + 1.64·37-s + 0.160·39-s + 0.312·41-s + 1.21·43-s + 0.291·47-s − 3/7·49-s + 0.840·51-s − 0.274·53-s − 0.264·57-s − 1.30·59-s − 0.768·61-s + 0.251·63-s + 1.71·67-s − 1.66·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.59277405313786, −13.95092525285926, −13.50286244753472, −12.93953273711211, −12.56813306698867, −11.89656125127698, −11.40547382077957, −11.00541088774435, −10.65066840244295, −10.01351160486176, −9.231177200768244, −9.130916336744454, −8.069963099198574, −7.888514130803338, −7.275139594489099, −6.612723875302907, −6.103712805412964, −5.504183467731681, −4.913580734433307, −4.453457334748381, −3.982939067753974, −2.927208519833938, −2.411940160363610, −1.691066958068461, −0.8692202984185527, 0,
0.8692202984185527, 1.691066958068461, 2.411940160363610, 2.927208519833938, 3.982939067753974, 4.453457334748381, 4.913580734433307, 5.504183467731681, 6.103712805412964, 6.612723875302907, 7.275139594489099, 7.888514130803338, 8.069963099198574, 9.130916336744454, 9.231177200768244, 10.01351160486176, 10.65066840244295, 11.00541088774435, 11.40547382077957, 11.89656125127698, 12.56813306698867, 12.93953273711211, 13.50286244753472, 13.95092525285926, 14.59277405313786