L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s − 13-s − 15-s − 3·17-s + 6·19-s − 21-s + 5·23-s + 25-s + 27-s + 4·31-s + 33-s + 35-s + 5·37-s − 39-s + 7·41-s − 6·43-s − 45-s − 8·47-s − 6·49-s − 3·51-s − 3·53-s − 55-s + 6·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.258·15-s − 0.727·17-s + 1.37·19-s − 0.218·21-s + 1.04·23-s + 1/5·25-s + 0.192·27-s + 0.718·31-s + 0.174·33-s + 0.169·35-s + 0.821·37-s − 0.160·39-s + 1.09·41-s − 0.914·43-s − 0.149·45-s − 1.16·47-s − 6/7·49-s − 0.420·51-s − 0.412·53-s − 0.134·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.172724801\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.172724801\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−7.84272386146327183989330230060, −7.56553886203355986104980728904, −6.65842374887445217164159513212, −6.12049906907317504556157849934, −4.89481319901987061135136693500, −4.53263979460277499968306364873, −3.31776943937383018837982055262, −3.07047064958484649640009662140, −1.88341202589429120518934282367, −0.75287972009993143137079034881,
0.75287972009993143137079034881, 1.88341202589429120518934282367, 3.07047064958484649640009662140, 3.31776943937383018837982055262, 4.53263979460277499968306364873, 4.89481319901987061135136693500, 6.12049906907317504556157849934, 6.65842374887445217164159513212, 7.56553886203355986104980728904, 7.84272386146327183989330230060