| L(s) = 1 | + i·2-s + 1.49i·3-s − 4-s + (−1.69 − 1.45i)5-s − 1.49·6-s + 1.01i·7-s − i·8-s + 0.763·9-s + (1.45 − 1.69i)10-s + 2.42·11-s − 1.49i·12-s − 1.01·14-s + (2.18 − 2.53i)15-s + 16-s + 5.04i·17-s + 0.763i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.863i·3-s − 0.5·4-s + (−0.757 − 0.652i)5-s − 0.610·6-s + 0.385i·7-s − 0.353i·8-s + 0.254·9-s + (0.461 − 0.535i)10-s + 0.729·11-s − 0.431i·12-s − 0.272·14-s + (0.563 − 0.653i)15-s + 0.250·16-s + 1.22i·17-s + 0.180i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4912537744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4912537744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (1.69 + 1.45i)T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.49iT - 3T^{2} \) |
| 7 | \( 1 - 1.01iT - 7T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 17 | \( 1 - 5.04iT - 17T^{2} \) |
| 19 | \( 1 + 7.36T + 19T^{2} \) |
| 23 | \( 1 + 4.52iT - 23T^{2} \) |
| 29 | \( 1 + 6.38T + 29T^{2} \) |
| 31 | \( 1 + 6.28T + 31T^{2} \) |
| 37 | \( 1 - 1.80iT - 37T^{2} \) |
| 41 | \( 1 + 0.399T + 41T^{2} \) |
| 43 | \( 1 - 8.41iT - 43T^{2} \) |
| 47 | \( 1 + 0.146iT - 47T^{2} \) |
| 53 | \( 1 + 1.52iT - 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + 6.59iT - 67T^{2} \) |
| 71 | \( 1 - 1.36T + 71T^{2} \) |
| 73 | \( 1 + 15.9iT - 73T^{2} \) |
| 79 | \( 1 - 3.53T + 79T^{2} \) |
| 83 | \( 1 - 4.64iT - 83T^{2} \) |
| 89 | \( 1 + 5.60T + 89T^{2} \) |
| 97 | \( 1 + 8.27iT - 97T^{2} \) |
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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
−9.534366956000678310432905428332, −8.972883086352180615382635069215, −8.397966332829731539396289565706, −7.57896668720821802056295366251, −6.57003196420791805653355259464, −5.82025497326549022149920467096, −4.72468432697765980946626063671, −4.21531313909418500536290223623, −3.54446372833235716174881048350, −1.68770337498658150328832292909,
0.18908076105386363095694728043, 1.56492364036353746374183372966, 2.56059189129752786938534202269, 3.77691682623651383174333880747, 4.25449560750489632717238575468, 5.61080531135369503136832878613, 6.76560078991702648825383471223, 7.20617923137052566143294914469, 7.908768380394870147741874132044, 8.912557912206522126415312006244
