L(s) = 1 | − 3.14i·5-s − 3.44i·7-s + 4.56·11-s + 6.89·13-s − 3.46i·17-s + 4.89i·19-s + 2.82·23-s − 4.89·25-s + 2.19i·29-s + 2.55i·31-s − 10.8·35-s + 4.89·37-s − 4.09i·41-s + 2.89i·43-s − 2.19·47-s + ⋯ |
L(s) = 1 | − 1.40i·5-s − 1.30i·7-s + 1.37·11-s + 1.91·13-s − 0.840i·17-s + 1.12i·19-s + 0.589·23-s − 0.979·25-s + 0.407i·29-s + 0.458i·31-s − 1.83·35-s + 0.805·37-s − 0.640i·41-s + 0.442i·43-s − 0.319·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.088824475\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.088824475\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.14iT - 5T^{2} \) |
| 7 | \( 1 + 3.44iT - 7T^{2} \) |
| 11 | \( 1 - 4.56T + 11T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 2.19iT - 29T^{2} \) |
| 31 | \( 1 - 2.55iT - 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 + 4.09iT - 41T^{2} \) |
| 43 | \( 1 - 2.89iT - 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 + 12.9iT - 53T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 14.8iT - 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 7.89T + 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 5.02iT - 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−8.954046958421142584365237174992, −8.548242060232286621352275671119, −7.60553408858924399893568300893, −6.69667638843024796739733118336, −5.92349457711189599691719064501, −4.87259611764327804054519821529, −4.04001557932732093333998421094, −3.52789201108675002635632690170, −1.38662198568049194703924992363, −0.996301559503478775598387172555,
1.49626030450248948723206705164, 2.70330996327922190236738881521, 3.44895974384156035646219924001, 4.39806917558924653224754464833, 5.96328275883850877402991009919, 6.16304633269776163795266460657, 6.91407422916258423740548275002, 8.001743375097391299772455144062, 8.947769710190805751367003433613, 9.226834688813224981327255444583