| L(s) = 1 | − 2.72i·2-s − 2.29i·3-s − 5.41·4-s − 6.24·6-s − 3.82i·7-s + 9.30i·8-s − 2.25·9-s − 4.41·11-s + 12.4i·12-s − 3.67i·13-s − 10.4·14-s + 14.5·16-s + 2.28i·17-s + 6.14i·18-s + 2.39·19-s + ⋯ |
| L(s) = 1 | − 1.92i·2-s − 1.32i·3-s − 2.70·4-s − 2.54·6-s − 1.44i·7-s + 3.29i·8-s − 0.752·9-s − 1.33·11-s + 3.58i·12-s − 1.01i·13-s − 2.78·14-s + 3.62·16-s + 0.554i·17-s + 1.44i·18-s + 0.548·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.821684 + 0.193973i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.821684 + 0.193973i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
| good | 2 | \( 1 + 2.72iT - 2T^{2} \) |
| 3 | \( 1 + 2.29iT - 3T^{2} \) |
| 7 | \( 1 + 3.82iT - 7T^{2} \) |
| 11 | \( 1 + 4.41T + 11T^{2} \) |
| 13 | \( 1 + 3.67iT - 13T^{2} \) |
| 17 | \( 1 - 2.28iT - 17T^{2} \) |
| 19 | \( 1 - 2.39T + 19T^{2} \) |
| 23 | \( 1 + 0.265iT - 23T^{2} \) |
| 29 | \( 1 - 6.58T + 29T^{2} \) |
| 31 | \( 1 - 2.34T + 31T^{2} \) |
| 41 | \( 1 + 4.41T + 41T^{2} \) |
| 43 | \( 1 - 7.71iT - 43T^{2} \) |
| 47 | \( 1 + 10.9iT - 47T^{2} \) |
| 53 | \( 1 + 0.109iT - 53T^{2} \) |
| 59 | \( 1 - 2.00T + 59T^{2} \) |
| 61 | \( 1 - 3.96T + 61T^{2} \) |
| 67 | \( 1 + 6.80iT - 67T^{2} \) |
| 71 | \( 1 + 5.79T + 71T^{2} \) |
| 73 | \( 1 + 0.140iT - 73T^{2} \) |
| 79 | \( 1 - 6.62T + 79T^{2} \) |
| 83 | \( 1 - 13.9iT - 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 8.94iT - 97T^{2} \) |
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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
−9.829694210046382603898277956703, −8.239168096882278608465822444910, −8.030969592806617863009882975917, −6.94374080227281110587386499589, −5.51007114512561863550633796083, −4.51771145598034684098831062759, −3.37677970202854347661616109158, −2.51078558619050788589290600863, −1.29673367807400129306463782690, −0.44796225027681243930166302800,
2.93250636459687595591788338551, 4.29084205143610601863771950725, 5.05429571463933879766379714700, 5.49072049436400653790090384087, 6.46091643411655891206517040749, 7.47621647508788617498020686339, 8.455043728017568169232643743406, 8.966765607356632013032481307155, 9.661383485057328941945804107749, 10.31892971211019705342679962177
