L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 5-s + (−1.78 − 3.08i)7-s − 0.999·8-s + (−0.5 + 0.866i)10-s + (−2.06 + 3.57i)11-s + (−3.34 − 1.35i)13-s − 3.56·14-s + (−0.5 + 0.866i)16-s + (2.56 + 4.43i)17-s + (1.78 + 3.08i)19-s + (0.499 + 0.866i)20-s + (2.06 + 3.57i)22-s + (−3.84 + 6.65i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (−0.673 − 1.16i)7-s − 0.353·8-s + (−0.158 + 0.273i)10-s + (−0.621 + 1.07i)11-s + (−0.926 − 0.375i)13-s − 0.951·14-s + (−0.125 + 0.216i)16-s + (0.621 + 1.07i)17-s + (0.408 + 0.707i)19-s + (0.111 + 0.193i)20-s + (0.439 + 0.761i)22-s + (−0.801 + 1.38i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4562775276\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4562775276\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + (3.34 + 1.35i)T \) |
good | 7 | \( 1 + (1.78 + 3.08i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.06 - 3.57i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.56 - 4.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.78 - 3.08i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.84 - 6.65i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.28 + 5.68i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 + (2.06 - 3.57i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.12 - 3.67i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.28 + 3.95i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 7T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 + (-5.28 - 9.14i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.12 - 12.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.43 + 4.22i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 7.43T + 79T^{2} \) |
| 83 | \( 1 + 1.12T + 83T^{2} \) |
| 89 | \( 1 + (-0.903 + 1.56i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.561 + 0.972i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.16005257660228774141216839918, −9.649278582196758825981679064032, −8.074337186320830070563668062788, −7.62417105331064707958935710392, −6.66728272190647394286437219377, −5.59013702407048418358423884709, −4.55822160272687823846799248988, −3.81770032688827370200436619814, −2.90947819592377417424378958648, −1.48234768449722870925741437243,
0.17336768521322871030647396929, 2.66605665155458832888053097825, 3.17559784853414327900779061942, 4.67313871087248402472005420313, 5.31184017457850902091929999632, 6.24839984934933855310698617700, 6.98598385497178588405945249553, 7.968188571998846357872101841031, 8.665270233846152077282399999969, 9.389229176577256544261690626565