L(s) = 1 | − 2-s + 1.41·3-s + 4-s − 1.41·5-s − 1.41·6-s − 8-s − 0.999·9-s + 1.41·10-s − 11-s + 1.41·12-s − 2.82·13-s − 2.00·15-s + 16-s + 5.65·17-s + 0.999·18-s − 2.82·19-s − 1.41·20-s + 22-s − 2·23-s − 1.41·24-s − 2.99·25-s + 2.82·26-s − 5.65·27-s − 6·29-s + 2.00·30-s + 1.41·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.816·3-s + 0.5·4-s − 0.632·5-s − 0.577·6-s − 0.353·8-s − 0.333·9-s + 0.447·10-s − 0.301·11-s + 0.408·12-s − 0.784·13-s − 0.516·15-s + 0.250·16-s + 1.37·17-s + 0.235·18-s − 0.648·19-s − 0.316·20-s + 0.213·22-s − 0.417·23-s − 0.288·24-s − 0.599·25-s + 0.554·26-s − 1.08·27-s − 1.11·29-s + 0.365·30-s + 0.254·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−9.450547251815216094657338083391, −8.540367026777806779843138756859, −7.84082279963447650020500724430, −7.47999686336930545781229172891, −6.21031904684137313513723697830, −5.18969643608982086215754502890, −3.83061144755951801039264188141, −2.99381486421233514984700654506, −1.89920185192110653583692761736, 0,
1.89920185192110653583692761736, 2.99381486421233514984700654506, 3.83061144755951801039264188141, 5.18969643608982086215754502890, 6.21031904684137313513723697830, 7.47999686336930545781229172891, 7.84082279963447650020500724430, 8.540367026777806779843138756859, 9.450547251815216094657338083391