L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.999·6-s + (−2.58 + 4.47i)7-s − 0.999·8-s + (1 + 1.73i)9-s − 0.999·10-s − 1.16·11-s + (0.499 + 0.866i)12-s + (2.08 − 3.60i)13-s − 5.16·14-s + (0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.581 − 1.00i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.408·6-s + (−0.975 + 1.68i)7-s − 0.353·8-s + (0.333 + 0.577i)9-s − 0.316·10-s − 0.350·11-s + (0.144 + 0.249i)12-s + (0.577 − 0.999i)13-s − 1.37·14-s + (0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.140 − 0.244i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.771133 + 1.16275i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.771133 + 1.16275i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (6.08 - 0.140i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (2.58 - 4.47i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 + (-2.08 + 3.60i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.581 + 1.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.16 - 7.20i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 - 4.16T + 31T^{2} \) |
| 41 | \( 1 + (2.66 - 4.61i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 7.32T + 43T^{2} \) |
| 47 | \( 1 - 5.16T + 47T^{2} \) |
| 53 | \( 1 + (7.08 + 12.2i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.581 - 1.00i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.16 + 7.20i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.16 + 7.20i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.16 - 5.47i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 3.48T + 73T^{2} \) |
| 79 | \( 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.32 - 14.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−12.16027824726497892574224771079, −10.72199561467672104050707787259, −9.782281948823175082458951043221, −8.415203042153325915714280403408, −8.159175167445902562343513687573, −6.75523878169826539666907379096, −6.06187629044709327909016780517, −5.06752791065000670970333675116, −3.37243343721468402506650079847, −2.41594353827370523126456088029,
0.846004990505883787091900343162, 3.00546221648908449843011707036, 4.13064452592503863477750881196, 4.53189737304267864144303863949, 6.45932592562941152013652533914, 7.07530387084052659853316631830, 8.718888934810170215192977108381, 9.371074288863706119755890118952, 10.42167334909846941011005694818, 10.82074690195222276451661557181