L(s) = 1 | + 0.339i·3-s + 3.88·7-s + 2.88·9-s − 1.54i·11-s + (−0.660 − 3.54i)13-s + 2.86i·19-s + 1.32i·21-s − 5.42i·23-s + 2i·27-s + 5.20·29-s − 6.22i·31-s + 0.524·33-s − 8.56·37-s + (1.20 − 0.224i)39-s + 9.08i·41-s + ⋯ |
L(s) = 1 | + 0.196i·3-s + 1.46·7-s + 0.961·9-s − 0.465i·11-s + (−0.183 − 0.983i)13-s + 0.657i·19-s + 0.288i·21-s − 1.13i·23-s + 0.384i·27-s + 0.966·29-s − 1.11i·31-s + 0.0913·33-s − 1.40·37-s + (0.192 − 0.0359i)39-s + 1.41i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.105990378\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.105990378\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (0.660 + 3.54i)T \) |
good | 3 | \( 1 - 0.339iT - 3T^{2} \) |
| 7 | \( 1 - 3.88T + 7T^{2} \) |
| 11 | \( 1 + 1.54iT - 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 2.86iT - 19T^{2} \) |
| 23 | \( 1 + 5.42iT - 23T^{2} \) |
| 29 | \( 1 - 5.20T + 29T^{2} \) |
| 31 | \( 1 + 6.22iT - 31T^{2} \) |
| 37 | \( 1 + 8.56T + 37T^{2} \) |
| 41 | \( 1 - 9.08iT - 41T^{2} \) |
| 43 | \( 1 + 0.980iT - 43T^{2} \) |
| 47 | \( 1 + 6.52T + 47T^{2} \) |
| 53 | \( 1 + 6.44iT - 53T^{2} \) |
| 59 | \( 1 - 4.45iT - 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 - 6.97T + 67T^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 - 3.43T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 8.56T + 83T^{2} \) |
| 89 | \( 1 + 17.1iT - 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−9.850830263715651207515544811593, −8.494108694458214852886665469711, −8.174059971775701215755482900113, −7.30108202350398069223513215236, −6.28255240534509867243023849184, −5.20066851066008326175259463985, −4.61453352908165580185550950457, −3.59827462987004916157621817741, −2.24704616834037225355164745271, −1.05399794786229604523363617308,
1.38700541683735393516857829277, 2.11677515441819475717118098002, 3.73825496080945422225693934581, 4.73100533158260182350020353396, 5.17802971479965389165005469833, 6.68036515383072158708121525818, 7.18961004544745033772543716226, 8.011939750472846202920500540259, 8.846361860490412467081635522100, 9.652833345869602705329307863661