L(s) = 1 | + i·2-s + (0.215 + 0.215i)3-s − 4-s + (0.336 − 2.21i)5-s + (−0.215 + 0.215i)6-s + (−0.673 − 0.673i)7-s − i·8-s − 2.90i·9-s + (2.21 + 0.336i)10-s − 1.69i·11-s + (−0.215 − 0.215i)12-s − 2.06i·13-s + (0.673 − 0.673i)14-s + (0.548 − 0.403i)15-s + 16-s + 2.58·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.124 + 0.124i)3-s − 0.5·4-s + (0.150 − 0.988i)5-s + (−0.0878 + 0.0878i)6-s + (−0.254 − 0.254i)7-s − 0.353i·8-s − 0.969i·9-s + (0.699 + 0.106i)10-s − 0.512i·11-s + (−0.0621 − 0.0621i)12-s − 0.571i·13-s + (0.179 − 0.179i)14-s + (0.141 − 0.104i)15-s + 0.250·16-s + 0.625·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19546 - 0.331567i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19546 - 0.331567i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (-0.336 + 2.21i)T \) |
| 37 | \( 1 + (-5.90 - 1.45i)T \) |
good | 3 | \( 1 + (-0.215 - 0.215i)T + 3iT^{2} \) |
| 7 | \( 1 + (0.673 + 0.673i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.69iT - 11T^{2} \) |
| 13 | \( 1 + 2.06iT - 13T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 + (0.312 + 0.312i)T + 19iT^{2} \) |
| 23 | \( 1 - 3.40iT - 23T^{2} \) |
| 29 | \( 1 + (-2.85 + 2.85i)T - 29iT^{2} \) |
| 31 | \( 1 + (0.382 + 0.382i)T + 31iT^{2} \) |
| 41 | \( 1 - 3.11iT - 41T^{2} \) |
| 43 | \( 1 - 7.38iT - 43T^{2} \) |
| 47 | \( 1 + (-1.26 - 1.26i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.69 - 2.69i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3.61 - 3.61i)T + 59iT^{2} \) |
| 61 | \( 1 + (4.54 + 4.54i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8.19 + 8.19i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.12T + 71T^{2} \) |
| 73 | \( 1 + (-3.81 - 3.81i)T + 73iT^{2} \) |
| 79 | \( 1 + (3.33 + 3.33i)T + 79iT^{2} \) |
| 83 | \( 1 + (6.48 - 6.48i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.486 - 0.486i)T - 89iT^{2} \) |
| 97 | \( 1 - 0.430T + 97T^{2} \) |
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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
−11.44305516889663357489214132758, −9.993780616711061160459800004558, −9.426687083318991865757101783066, −8.471217760220782995410297645597, −7.70449693262569493758601870526, −6.37799414574385367022228959388, −5.61581064692847615914082863175, −4.44446496079738331055840489169, −3.30457411713041323180116144858, −0.880551271798739073842238232886,
1.99620226903074505290369470278, 2.95856124031334565764723489765, 4.30839935129956202510573597066, 5.57967772947146728396997492523, 6.80809926112869128609334019063, 7.73411140819758395409533974446, 8.854992802781847509474465848507, 9.937779997003719780903606826643, 10.53639025212512101655074353897, 11.37919022603785936344495108117