L(s) = 1 | − 2.48i·2-s + 1.67·3-s − 4.15·4-s + 0.675i·5-s − 4.15i·6-s − i·7-s + 5.35i·8-s − 0.193·9-s + 1.67·10-s + 4.48i·11-s − 6.96·12-s + (3.28 + 1.48i)13-s − 2.48·14-s + 1.13i·15-s + 4.96·16-s − 3.28·17-s + ⋯ |
L(s) = 1 | − 1.75i·2-s + 0.967·3-s − 2.07·4-s + 0.301i·5-s − 1.69i·6-s − 0.377i·7-s + 1.89i·8-s − 0.0646·9-s + 0.529·10-s + 1.35i·11-s − 2.00·12-s + (0.911 + 0.410i)13-s − 0.663·14-s + 0.292i·15-s + 1.24·16-s − 0.797·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.615013 - 0.951679i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.615013 - 0.951679i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + iT \) |
| 13 | \( 1 + (-3.28 - 1.48i)T \) |
good | 2 | \( 1 + 2.48iT - 2T^{2} \) |
| 3 | \( 1 - 1.67T + 3T^{2} \) |
| 5 | \( 1 - 0.675iT - 5T^{2} \) |
| 11 | \( 1 - 4.48iT - 11T^{2} \) |
| 17 | \( 1 + 3.28T + 17T^{2} \) |
| 19 | \( 1 + 5.21iT - 19T^{2} \) |
| 23 | \( 1 - 4.76T + 23T^{2} \) |
| 29 | \( 1 + 9.31T + 29T^{2} \) |
| 31 | \( 1 - 1.63iT - 31T^{2} \) |
| 37 | \( 1 + 1.44iT - 37T^{2} \) |
| 41 | \( 1 + 7.92iT - 41T^{2} \) |
| 43 | \( 1 + 4.61T + 43T^{2} \) |
| 47 | \( 1 - 7.86iT - 47T^{2} \) |
| 53 | \( 1 + 3.15T + 53T^{2} \) |
| 59 | \( 1 + 2.54iT - 59T^{2} \) |
| 61 | \( 1 + 2.31T + 61T^{2} \) |
| 67 | \( 1 - 7.35iT - 67T^{2} \) |
| 71 | \( 1 - 7.75iT - 71T^{2} \) |
| 73 | \( 1 + 15.1iT - 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 1.45iT - 83T^{2} \) |
| 89 | \( 1 + 7.79iT - 89T^{2} \) |
| 97 | \( 1 + 17.9iT - 97T^{2} \) |
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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
−13.41864048270695316227915456715, −12.81393082670946613469479312458, −11.38545309976043029653588784944, −10.73038186074594769689269652180, −9.383600005778452324614398126001, −8.841679912133318356206106547534, −7.13783554861322453754111723515, −4.60458169138919007256741610810, −3.34558627209866481755641184788, −2.06041210560032184859029994599,
3.53525046288720474932622155674, 5.37440888506246759159074062061, 6.35649436484857851357481458100, 7.926224027525450607697817132418, 8.571885557009975947464026812881, 9.233992423511432531015367765288, 11.14806387964268816168984566564, 13.10268496781018272864341576876, 13.64424879761094911846614659756, 14.70246133465779420293577137426