| L(s) = 1 | + 2.67·3-s − 4.38·7-s + 4.16·9-s − 11-s + 4·13-s + 5.87·17-s + 0.526·19-s − 11.7·21-s − 6.15·23-s + 3.11·27-s − 0.967·29-s + 9.60·31-s − 2.67·33-s + 1.76·37-s + 10.7·39-s + 2.50·41-s + 3.85·43-s + 4.91·47-s + 12.2·49-s + 15.7·51-s − 5.29·53-s + 1.40·57-s + 2.63·59-s + 8.96·61-s − 18.2·63-s + 7.97·67-s − 16.4·69-s + ⋯ |
| L(s) = 1 | + 1.54·3-s − 1.65·7-s + 1.38·9-s − 0.301·11-s + 1.10·13-s + 1.42·17-s + 0.120·19-s − 2.56·21-s − 1.28·23-s + 0.600·27-s − 0.179·29-s + 1.72·31-s − 0.465·33-s + 0.290·37-s + 1.71·39-s + 0.391·41-s + 0.588·43-s + 0.716·47-s + 1.74·49-s + 2.20·51-s − 0.727·53-s + 0.186·57-s + 0.343·59-s + 1.14·61-s − 2.30·63-s + 0.974·67-s − 1.98·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.032840842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.032840842\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2.67T + 3T^{2} \) |
| 7 | \( 1 + 4.38T + 7T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 5.87T + 17T^{2} \) |
| 19 | \( 1 - 0.526T + 19T^{2} \) |
| 23 | \( 1 + 6.15T + 23T^{2} \) |
| 29 | \( 1 + 0.967T + 29T^{2} \) |
| 31 | \( 1 - 9.60T + 31T^{2} \) |
| 37 | \( 1 - 1.76T + 37T^{2} \) |
| 41 | \( 1 - 2.50T + 41T^{2} \) |
| 43 | \( 1 - 3.85T + 43T^{2} \) |
| 47 | \( 1 - 4.91T + 47T^{2} \) |
| 53 | \( 1 + 5.29T + 53T^{2} \) |
| 59 | \( 1 - 2.63T + 59T^{2} \) |
| 61 | \( 1 - 8.96T + 61T^{2} \) |
| 67 | \( 1 - 7.97T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 7.61T + 83T^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.176650154259592865369259605350, −7.983179216072711570391561820128, −6.98971132062605656401489592102, −6.23704631504350739632551143339, −5.60838322695073769837183723797, −4.16985064083593498307198440919, −3.57354501186573332574672526649, −3.02957665983873617792127097656, −2.26618436351945328672215115281, −0.916360201751005595802340589951,
0.916360201751005595802340589951, 2.26618436351945328672215115281, 3.02957665983873617792127097656, 3.57354501186573332574672526649, 4.16985064083593498307198440919, 5.60838322695073769837183723797, 6.23704631504350739632551143339, 6.98971132062605656401489592102, 7.983179216072711570391561820128, 8.176650154259592865369259605350
