L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.71 + 0.208i)3-s + 1.00i·4-s + (−0.313 + 1.16i)5-s + (−1.06 − 1.36i)6-s + (−1.16 + 4.35i)7-s + (0.707 − 0.707i)8-s + (2.91 + 0.715i)9-s + (1.04 − 0.604i)10-s + (0.617 − 0.617i)11-s + (−0.208 + 1.71i)12-s + (−1.15 + 3.41i)13-s + (3.90 − 2.25i)14-s + (−0.781 + 1.94i)15-s − 1.00·16-s + (1.49 − 2.58i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.992 + 0.120i)3-s + 0.500i·4-s + (−0.140 + 0.522i)5-s + (−0.436 − 0.556i)6-s + (−0.440 + 1.64i)7-s + (0.250 − 0.250i)8-s + (0.971 + 0.238i)9-s + (0.331 − 0.191i)10-s + (0.186 − 0.186i)11-s + (−0.0600 + 0.496i)12-s + (−0.319 + 0.947i)13-s + (1.04 − 0.601i)14-s + (−0.201 + 0.501i)15-s − 0.250·16-s + (0.362 − 0.627i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19604 + 0.332475i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19604 + 0.332475i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-1.71 - 0.208i)T \) |
| 13 | \( 1 + (1.15 - 3.41i)T \) |
good | 5 | \( 1 + (0.313 - 1.16i)T + (-4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (1.16 - 4.35i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.617 + 0.617i)T - 11iT^{2} \) |
| 17 | \( 1 + (-1.49 + 2.58i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.92 + 7.17i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (0.0774 - 0.134i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 9.39iT - 29T^{2} \) |
| 31 | \( 1 + (-4.22 - 1.13i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (1.18 - 4.40i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.46 + 1.46i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-2.01 + 1.16i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.69 + 6.31i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 - 0.732iT - 53T^{2} \) |
| 59 | \( 1 + (10.1 - 10.1i)T - 59iT^{2} \) |
| 61 | \( 1 + (1.64 + 2.84i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.37 + 5.14i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (14.6 - 3.93i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.594 - 0.594i)T + 73iT^{2} \) |
| 79 | \( 1 + (-1.78 + 3.09i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.44 - 0.654i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-5.91 - 1.58i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-10.7 - 2.88i)T + (84.0 + 48.5i)T^{2} \) |
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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
−12.07261213691486494490424132674, −11.40394024201952440235026449232, −10.07711166915891885150854198746, −9.139573227441576439439250786895, −8.801088805626011395856915637385, −7.47671068393247993469057180272, −6.41086524149821699014791172340, −4.62351729522225911608566237766, −3.02050679193811552519052151666, −2.33765028676359115959961147396,
1.23498551683818153093438382588, 3.43651169819542145907671473995, 4.53959682072251135796270153116, 6.28659834953122782469788801076, 7.46675049138682367715152355119, 7.972675586281123793526402001166, 9.037846456845746074568351396763, 10.18406453884093406156433136333, 10.51765214217612687691622794634, 12.53994739410256633066096289592