| L(s) = 1 | + 3.23i·3-s − 1.23i·5-s − 7-s − 7.47·9-s + 2.47i·11-s − 5.23i·13-s + 4.00·15-s − 4.47·17-s − 3.23i·19-s − 3.23i·21-s − 4·23-s + 3.47·25-s − 14.4i·27-s − 4.47i·29-s + 6.47·31-s + ⋯ |
| L(s) = 1 | + 1.86i·3-s − 0.552i·5-s − 0.377·7-s − 2.49·9-s + 0.745i·11-s − 1.45i·13-s + 1.03·15-s − 1.08·17-s − 0.742i·19-s − 0.706i·21-s − 0.834·23-s + 0.694·25-s − 2.78i·27-s − 0.830i·29-s + 1.16·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7030403323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7030403323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 3.23iT - 3T^{2} \) |
| 5 | \( 1 + 1.23iT - 5T^{2} \) |
| 11 | \( 1 - 2.47iT - 11T^{2} \) |
| 13 | \( 1 + 5.23iT - 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 3.23iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4.47iT - 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47iT - 37T^{2} \) |
| 41 | \( 1 + 0.472T + 41T^{2} \) |
| 43 | \( 1 + 2.47iT - 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 4.76iT - 59T^{2} \) |
| 61 | \( 1 + 6.76iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 - 4.76iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−9.271142011246617150926849788876, −8.640476977756705220412920397117, −7.942220728498281929499496741176, −6.59021682681677904205425769295, −5.67992878438761921294310125418, −4.81681060798973566215822388612, −4.42802451660718185468943516494, −3.37537502221229964077450199679, −2.51481100218841815462307982856, −0.26549596608519342131853183241,
1.28263671569330818281342164941, 2.28038867430418944832058444007, 3.09007272737051820480564957414, 4.36586457338485032119634414987, 5.91798504323521121380142921577, 6.26391838175745026487182060941, 7.03139480891636410492251369027, 7.51913463535147148042611393246, 8.665579463792887300864268011274, 8.896879393632613712401302801960
