| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 4.89i·7-s + i·8-s − 9-s − i·12-s + 2.89i·13-s + 4.89·14-s + 16-s − i·17-s + i·18-s − 4·19-s − 4.89·21-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.85i·7-s + 0.353i·8-s − 0.333·9-s − 0.288i·12-s + 0.804i·13-s + 1.30·14-s + 0.250·16-s − 0.242i·17-s + 0.235i·18-s − 0.917·19-s − 1.06·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7753025667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7753025667\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 - 4.89iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2.89iT - 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 6.89T + 41T^{2} \) |
| 43 | \( 1 - 0.898iT - 43T^{2} \) |
| 47 | \( 1 + 9.79iT - 47T^{2} \) |
| 53 | \( 1 + 11.7iT - 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 + 7.79T + 61T^{2} \) |
| 67 | \( 1 + 8.89iT - 67T^{2} \) |
| 71 | \( 1 - 0.898T + 71T^{2} \) |
| 73 | \( 1 - 1.10iT - 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 5.79iT - 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 16.6iT - 97T^{2} \) |
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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
−9.383491671912415463759724948106, −8.655318112539180948485352828815, −8.120910477083480910603916572807, −6.78527493986793413657171787058, −5.87987691284628707246006574945, −5.25227805831730867440728803842, −4.42950169149649833523900349573, −3.46063575178426086665261378584, −2.52881151657490321322747068675, −1.80624680830123940465618460201,
0.26362376804872459902107068556, 1.29040103228238986982684350441, 2.80885593434393858770740299818, 4.01559655696733607028704117303, 4.46726294443081497500882844724, 5.71308281736953227703399701180, 6.36011126016323038697867435359, 7.21552098006507615737752315395, 7.60704981894494113255565636704, 8.288840830487324470528551343664
