L(s) = 1 | − i·2-s + (−0.0386 + 1.73i)3-s − 4-s + 5-s + (1.73 + 0.0386i)6-s + 2.83i·7-s + i·8-s + (−2.99 − 0.133i)9-s − i·10-s − 4.50·11-s + (0.0386 − 1.73i)12-s − 5.89·13-s + 2.83·14-s + (−0.0386 + 1.73i)15-s + 16-s + 5.40·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.0223 + 0.999i)3-s − 0.5·4-s + 0.447·5-s + (0.706 + 0.0157i)6-s + 1.07i·7-s + 0.353i·8-s + (−0.999 − 0.0446i)9-s − 0.316i·10-s − 1.35·11-s + (0.0111 − 0.499i)12-s − 1.63·13-s + 0.758·14-s + (−0.00998 + 0.447i)15-s + 0.250·16-s + 1.30·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.225420 + 0.588956i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.225420 + 0.588956i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.0386 - 1.73i)T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + (3.64 + 3.12i)T \) |
good | 7 | \( 1 - 2.83iT - 7T^{2} \) |
| 11 | \( 1 + 4.50T + 11T^{2} \) |
| 13 | \( 1 + 5.89T + 13T^{2} \) |
| 17 | \( 1 - 5.40T + 17T^{2} \) |
| 19 | \( 1 - 2.26iT - 19T^{2} \) |
| 29 | \( 1 + 4.81iT - 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 - 3.92iT - 37T^{2} \) |
| 41 | \( 1 + 2.24iT - 41T^{2} \) |
| 43 | \( 1 - 11.5iT - 43T^{2} \) |
| 47 | \( 1 - 1.55iT - 47T^{2} \) |
| 53 | \( 1 - 6.08T + 53T^{2} \) |
| 59 | \( 1 - 10.8iT - 59T^{2} \) |
| 61 | \( 1 - 11.6iT - 61T^{2} \) |
| 67 | \( 1 + 3.47iT - 67T^{2} \) |
| 71 | \( 1 - 9.84iT - 71T^{2} \) |
| 73 | \( 1 - 0.323T + 73T^{2} \) |
| 79 | \( 1 - 5.69iT - 79T^{2} \) |
| 83 | \( 1 - 4.31T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 8.21iT - 97T^{2} \) |
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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
−10.50810402348885999650101820869, −9.994148332463008512765335446526, −9.458978421200046574615786816555, −8.438350774852925202034550604896, −7.61489878010949071099431682953, −5.70794503542593401426907639993, −5.42707212835597925852726869429, −4.36959238814376917469283325230, −2.94313313867392928350955850437, −2.32720071861534730433796703249,
0.30689417752803153949179237453, 2.06670943907084468210127962646, 3.43848359274430728519729590121, 5.12405839549860028380665529969, 5.54717892792964770948358515722, 6.92122599588957632828915214724, 7.46237392481472286771560361771, 7.920117583736547157624776525174, 9.203988198525829288440454055307, 10.11047775071199021183347700956