L(s) = 1 | + 3.45·3-s + 2.04·5-s + 2.58·7-s + 8.93·9-s − 2.30·11-s − 4.87·13-s + 7.05·15-s + 17-s + 4.97·19-s + 8.93·21-s + 3.63·23-s − 0.832·25-s + 20.4·27-s + 8.15·29-s − 7.13·31-s − 7.95·33-s + 5.27·35-s + 5.94·37-s − 16.8·39-s + 0.901·41-s − 3.72·43-s + 18.2·45-s + 2.64·47-s − 0.312·49-s + 3.45·51-s − 13.9·53-s − 4.70·55-s + ⋯ |
L(s) = 1 | + 1.99·3-s + 0.912·5-s + 0.977·7-s + 2.97·9-s − 0.694·11-s − 1.35·13-s + 1.82·15-s + 0.242·17-s + 1.14·19-s + 1.94·21-s + 0.758·23-s − 0.166·25-s + 3.94·27-s + 1.51·29-s − 1.28·31-s − 1.38·33-s + 0.892·35-s + 0.976·37-s − 2.69·39-s + 0.140·41-s − 0.568·43-s + 2.71·45-s + 0.385·47-s − 0.0445·49-s + 0.483·51-s − 1.91·53-s − 0.634·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.305243680\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.305243680\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - 3.45T + 3T^{2} \) |
| 5 | \( 1 - 2.04T + 5T^{2} \) |
| 7 | \( 1 - 2.58T + 7T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 13 | \( 1 + 4.87T + 13T^{2} \) |
| 19 | \( 1 - 4.97T + 19T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 - 8.15T + 29T^{2} \) |
| 31 | \( 1 + 7.13T + 31T^{2} \) |
| 37 | \( 1 - 5.94T + 37T^{2} \) |
| 41 | \( 1 - 0.901T + 41T^{2} \) |
| 43 | \( 1 + 3.72T + 43T^{2} \) |
| 47 | \( 1 - 2.64T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 61 | \( 1 - 8.07T + 61T^{2} \) |
| 67 | \( 1 + 4.93T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 2.52T + 89T^{2} \) |
| 97 | \( 1 + 3.18T + 97T^{2} \) |
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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
−7.944084479283443707378154103302, −7.36991441840429499620087924086, −6.82734277449067989185411343900, −5.53908075305154544892275863319, −4.88961731027333857242399738768, −4.30998367837955922586605630439, −3.11531380212776818920212112806, −2.72034132395918145839973628117, −1.95093788878367569971755390962, −1.27359428686018079185178976303,
1.27359428686018079185178976303, 1.95093788878367569971755390962, 2.72034132395918145839973628117, 3.11531380212776818920212112806, 4.30998367837955922586605630439, 4.88961731027333857242399738768, 5.53908075305154544892275863319, 6.82734277449067989185411343900, 7.36991441840429499620087924086, 7.944084479283443707378154103302