L(s) = 1 | + (0.5 + 0.866i)2-s + (0.707 + 0.707i)3-s + (−0.499 + 0.866i)4-s + (−0.258 + 0.965i)6-s − 0.999·8-s + 1.00i·9-s − 1.73·11-s + (−0.965 + 0.258i)12-s + (−0.5 − 0.866i)16-s + (0.258 + 0.448i)17-s + (−0.866 + 0.500i)18-s + (−0.965 + 1.67i)19-s + (−0.866 − 1.49i)22-s + (−0.707 − 0.707i)24-s + 25-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (0.707 + 0.707i)3-s + (−0.499 + 0.866i)4-s + (−0.258 + 0.965i)6-s − 0.999·8-s + 1.00i·9-s − 1.73·11-s + (−0.965 + 0.258i)12-s + (−0.5 − 0.866i)16-s + (0.258 + 0.448i)17-s + (−0.866 + 0.500i)18-s + (−0.965 + 1.67i)19-s + (−0.866 − 1.49i)22-s + (−0.707 − 0.707i)24-s + 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.417645952\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.417645952\) |
\(L(1)\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−8.745238408206858040587345872091, −8.404519386551632394484201813271, −7.72920395365650266553315929440, −7.12054190596327903756991620436, −5.88450034569608601041087680584, −5.45228937303084828859561180865, −4.57508267526902485963812662902, −3.87526035425631563472533750549, −3.04154564988119759426859105337, −2.19487534438092709080542231404,
0.59682731085724558934234827916, 1.99290318564691387889049396951, 2.76236470512407913308326455598, 3.20360498733543133815547006951, 4.55813228784058033478003077797, 5.03130923314159004980148983519, 6.11055946683921448661024495027, 6.81329053999741369420666507326, 7.73307109725191233318615145917, 8.379514660505394694656986996312