| L(s) = 1 | − 4·7-s − 2.64i·11-s + 3·17-s + 2.64i·19-s + 4·23-s − 4·31-s + 10.5i·37-s + 5·41-s − 5.29i·43-s − 8·47-s + 9·49-s + 10.5i·53-s − 5.29i·59-s − 10.5i·61-s + 7.93i·67-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.797i·11-s + 0.727·17-s + 0.606i·19-s + 0.834·23-s − 0.718·31-s + 1.73i·37-s + 0.780·41-s − 0.806i·43-s − 1.16·47-s + 1.28·49-s + 1.45i·53-s − 0.688i·59-s − 1.35i·61-s + 0.969i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.155165278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155165278\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + 2.64iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 2.64iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 10.5iT - 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + 5.29iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 5.29iT - 59T^{2} \) |
| 61 | \( 1 + 10.5iT - 61T^{2} \) |
| 67 | \( 1 - 7.93iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 7.93iT - 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.76966906916801567575331181722, −7.00360196890716709492514941033, −6.35454903039550236384234717936, −5.82190375134004176654692491211, −5.07866921797401900035613302297, −4.00966874107604532919841206406, −3.24767074610921818382542598378, −2.88969913656101096342440400305, −1.50240995446539057150027547488, −0.37715829738033988477180786741,
0.794007711180575904483242653977, 2.09774223390790935898246191101, 2.97076495291055448433773075295, 3.60134864047200454574852803727, 4.44266778776969521428835948562, 5.33223594574185730910439578439, 5.98121948140712594472202619560, 6.81093829382915835528228498357, 7.17836075354556133763065438995, 7.945154055141996264059185395877
