L(s) = 1 | + 2-s + 4-s − 5-s − 4·7-s + 8-s − 10-s − 6·11-s + 2·13-s − 4·14-s + 16-s + 6·17-s + 2·19-s − 20-s − 6·22-s + 25-s + 2·26-s − 4·28-s − 6·29-s + 8·31-s + 32-s + 6·34-s + 4·35-s + 37-s + 2·38-s − 40-s + 6·41-s + 8·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s − 1.80·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s − 0.223·20-s − 1.27·22-s + 1/5·25-s + 0.392·26-s − 0.755·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.676·35-s + 0.164·37-s + 0.324·38-s − 0.158·40-s + 0.937·41-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.937164869\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.937164869\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.424912579227892857567266604331, −7.71588865532144785570929233268, −7.18162370286094383162742724296, −6.14700327202829166226347590269, −5.67324598464159891720762446042, −4.85568846993463928899190098959, −3.73420480399354799573660891311, −3.17995762137668904356640145254, −2.47135784523752124203055675221, −0.72114804858626419919238278961,
0.72114804858626419919238278961, 2.47135784523752124203055675221, 3.17995762137668904356640145254, 3.73420480399354799573660891311, 4.85568846993463928899190098959, 5.67324598464159891720762446042, 6.14700327202829166226347590269, 7.18162370286094383162742724296, 7.71588865532144785570929233268, 8.424912579227892857567266604331