L(s) = 1 | − 1.41·2-s + 3-s − 5-s − 1.41·6-s + 2.82·8-s + 9-s + 1.41·10-s − 2·11-s + 13-s − 15-s − 4.00·16-s + 1.41·17-s − 1.41·18-s − 3.58·19-s + 2.82·22-s + 0.414·23-s + 2.82·24-s − 4·25-s − 1.41·26-s + 27-s + 1.24·29-s + 1.41·30-s − 0.414·31-s − 2·33-s − 2.00·34-s + ⋯ |
L(s) = 1 | − 1.00·2-s + 0.577·3-s − 0.447·5-s − 0.577·6-s + 0.999·8-s + 0.333·9-s + 0.447·10-s − 0.603·11-s + 0.277·13-s − 0.258·15-s − 1.00·16-s + 0.342·17-s − 0.333·18-s − 0.822·19-s + 0.603·22-s + 0.0863·23-s + 0.577·24-s − 0.800·25-s − 0.277·26-s + 0.192·27-s + 0.230·29-s + 0.258·30-s − 0.0743·31-s − 0.348·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 23 | \( 1 - 0.414T + 23T^{2} \) |
| 29 | \( 1 - 1.24T + 29T^{2} \) |
| 31 | \( 1 + 0.414T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 - 3.82T + 43T^{2} \) |
| 47 | \( 1 + 4.65T + 47T^{2} \) |
| 53 | \( 1 - 2.41T + 53T^{2} \) |
| 59 | \( 1 - 7.65T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 4.58T + 67T^{2} \) |
| 71 | \( 1 + 6.82T + 71T^{2} \) |
| 73 | \( 1 + 5.58T + 73T^{2} \) |
| 79 | \( 1 - 1.48T + 79T^{2} \) |
| 83 | \( 1 + 5.48T + 83T^{2} \) |
| 89 | \( 1 - 2.65T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−8.802788799349662912668121023762, −8.074066595591203925246228321664, −7.66473641656575889689639890996, −6.78806105220811687161784593936, −5.62103796878376426609962436957, −4.53570714635406993888613817034, −3.81284491185912862588283291525, −2.61014193119911064757557484744, −1.44882733476222978687682703199, 0,
1.44882733476222978687682703199, 2.61014193119911064757557484744, 3.81284491185912862588283291525, 4.53570714635406993888613817034, 5.62103796878376426609962436957, 6.78806105220811687161784593936, 7.66473641656575889689639890996, 8.074066595591203925246228321664, 8.802788799349662912668121023762