L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 11-s + 4·13-s + 15-s − 0.298·17-s − 7.70·19-s + 21-s − 1.70·23-s + 25-s − 27-s + 9.10·29-s − 7.40·31-s − 33-s + 35-s − 5.40·37-s − 4·39-s + 7.40·41-s + 9.70·43-s − 45-s − 0.596·47-s + 49-s + 0.298·51-s − 7.10·53-s − 55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s + 0.301·11-s + 1.10·13-s + 0.258·15-s − 0.0723·17-s − 1.76·19-s + 0.218·21-s − 0.354·23-s + 0.200·25-s − 0.192·27-s + 1.69·29-s − 1.32·31-s − 0.174·33-s + 0.169·35-s − 0.888·37-s − 0.640·39-s + 1.15·41-s + 1.47·43-s − 0.149·45-s − 0.0870·47-s + 0.142·49-s + 0.0417·51-s − 0.975·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 0.298T + 17T^{2} \) |
| 19 | \( 1 + 7.70T + 19T^{2} \) |
| 23 | \( 1 + 1.70T + 23T^{2} \) |
| 29 | \( 1 - 9.10T + 29T^{2} \) |
| 31 | \( 1 + 7.40T + 31T^{2} \) |
| 37 | \( 1 + 5.40T + 37T^{2} \) |
| 41 | \( 1 - 7.40T + 41T^{2} \) |
| 43 | \( 1 - 9.70T + 43T^{2} \) |
| 47 | \( 1 + 0.596T + 47T^{2} \) |
| 53 | \( 1 + 7.10T + 53T^{2} \) |
| 59 | \( 1 + 9.10T + 59T^{2} \) |
| 61 | \( 1 + 0.298T + 61T^{2} \) |
| 67 | \( 1 + 0.596T + 67T^{2} \) |
| 71 | \( 1 + 3.40T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 + 0.298T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 4.29T + 97T^{2} \) |
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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
−7.968898407079492693520354321299, −7.13705088733036981003486101509, −6.26601771172209980669702222172, −6.09598021263219719871044863449, −4.90988512497545991759702426282, −4.15708704573121084487380558688, −3.56176382650058890821770194303, −2.38849530113585202697227337799, −1.22539964588816463736157985647, 0,
1.22539964588816463736157985647, 2.38849530113585202697227337799, 3.56176382650058890821770194303, 4.15708704573121084487380558688, 4.90988512497545991759702426282, 6.09598021263219719871044863449, 6.26601771172209980669702222172, 7.13705088733036981003486101509, 7.968898407079492693520354321299