L(s) = 1 | + 0.688i·2-s − 2.21·3-s + 1.52·4-s + 3.21i·5-s − 1.52i·6-s + i·7-s + 2.42i·8-s + 1.90·9-s − 2.21·10-s − 2.68i·11-s − 3.37·12-s + (3.59 + 0.311i)13-s − 0.688·14-s − 7.11i·15-s + 1.37·16-s − 3.59·17-s + ⋯ |
L(s) = 1 | + 0.487i·2-s − 1.27·3-s + 0.762·4-s + 1.43i·5-s − 0.622i·6-s + 0.377i·7-s + 0.858i·8-s + 0.634·9-s − 0.700·10-s − 0.810i·11-s − 0.975·12-s + (0.996 + 0.0862i)13-s − 0.184·14-s − 1.83i·15-s + 0.344·16-s − 0.871·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0862 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0862 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.594602 + 0.545330i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.594602 + 0.545330i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - iT \) |
| 13 | \( 1 + (-3.59 - 0.311i)T \) |
good | 2 | \( 1 - 0.688iT - 2T^{2} \) |
| 3 | \( 1 + 2.21T + 3T^{2} \) |
| 5 | \( 1 - 3.21iT - 5T^{2} \) |
| 11 | \( 1 + 2.68iT - 11T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 + 8.54iT - 19T^{2} \) |
| 23 | \( 1 - 3.28T + 23T^{2} \) |
| 29 | \( 1 - 2.05T + 29T^{2} \) |
| 31 | \( 1 - 5.83iT - 31T^{2} \) |
| 37 | \( 1 + 3.93iT - 37T^{2} \) |
| 41 | \( 1 - 0.755iT - 41T^{2} \) |
| 43 | \( 1 + 8.80T + 43T^{2} \) |
| 47 | \( 1 + 1.88iT - 47T^{2} \) |
| 53 | \( 1 - 2.52T + 53T^{2} \) |
| 59 | \( 1 + 7.33iT - 59T^{2} \) |
| 61 | \( 1 - 9.05T + 61T^{2} \) |
| 67 | \( 1 - 0.428iT - 67T^{2} \) |
| 71 | \( 1 - 8.98iT - 71T^{2} \) |
| 73 | \( 1 + 5.79iT - 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 10.8iT - 83T^{2} \) |
| 89 | \( 1 + 5.36iT - 89T^{2} \) |
| 97 | \( 1 - 9.62iT - 97T^{2} \) |
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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
−14.60120178530167868660386034700, −13.36418771109466447083758446791, −11.69373166903087370069160822320, −11.09273044815413333410432660964, −10.69031252975833843623176511870, −8.642346188122441345964472872470, −6.81074850700636120301994865983, −6.56026311914157551545955297033, −5.33654674742632997477513696031, −2.89522158791828787790384201645,
1.34673579296682142216681795402, 4.18756715711776749617986839205, 5.56018821179401057661646580728, 6.67216906210236000436164792559, 8.253813798608273811713691242534, 9.830863722000829546011011017194, 10.83035041745747949116012995888, 11.79235999144457625827927702558, 12.46352211811540891727289666759, 13.30544278620535514599108240396