| L(s) = 1 | + 2.37i·3-s + i·7-s − 2.62·9-s + 6.37·11-s + 4.37i·13-s + 0.372i·17-s + 4.74·19-s − 2.37·21-s − 4.74i·23-s + 0.883i·27-s + 4.37·29-s − 8·31-s + 15.1i·33-s + 2i·37-s − 10.3·39-s + ⋯ |
| L(s) = 1 | + 1.36i·3-s + 0.377i·7-s − 0.875·9-s + 1.92·11-s + 1.21i·13-s + 0.0902i·17-s + 1.08·19-s − 0.517·21-s − 0.989i·23-s + 0.169i·27-s + 0.811·29-s − 1.43·31-s + 2.63i·33-s + 0.328i·37-s − 1.66·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.874969935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.874969935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - 2.37iT - 3T^{2} \) |
| 11 | \( 1 - 6.37T + 11T^{2} \) |
| 13 | \( 1 - 4.37iT - 13T^{2} \) |
| 17 | \( 1 - 0.372iT - 17T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 23 | \( 1 + 4.74iT - 23T^{2} \) |
| 29 | \( 1 - 4.37T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6.74T + 41T^{2} \) |
| 43 | \( 1 + 8.74iT - 43T^{2} \) |
| 47 | \( 1 - 7.11iT - 47T^{2} \) |
| 53 | \( 1 - 10.7iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 2.74T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 9.48iT - 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 9.86iT - 97T^{2} \) |
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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
−9.537271975139311937829241812424, −9.226197626728592459924967430238, −8.681249870404914166408211031481, −7.29189226638239601667964196034, −6.47065943547911520903316357830, −5.59314833508952772011714162803, −4.44244714552740709788916182633, −4.09104890284744200222368086986, −3.04084918377768325476006952003, −1.51379402837320561826136960514,
0.878826684662716333999914239061, 1.63614467732179163470468212709, 3.07609754494843079586119052947, 3.98835724350761741555531399266, 5.36300106248411565105561122905, 6.19361746890550661782612452121, 6.98272888563043599793340005416, 7.52459587519560767090339135456, 8.294451067605778344873146695957, 9.285636565483264265757542536053
