| L(s) = 1 | + 1.80·2-s + 0.687·3-s + 1.26·4-s + 1.24·6-s − 4.34·7-s − 1.33·8-s − 2.52·9-s + 0.0525·11-s + 0.867·12-s − 3.02·13-s − 7.84·14-s − 4.93·16-s − 4.56·18-s + 7.82·19-s − 2.98·21-s + 0.0948·22-s + 1.04·23-s − 0.918·24-s − 5.46·26-s − 3.80·27-s − 5.47·28-s + 0.420·29-s − 1.38·31-s − 6.23·32-s + 0.0361·33-s − 3.18·36-s + 0.336·37-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 0.397·3-s + 0.630·4-s + 0.507·6-s − 1.64·7-s − 0.471·8-s − 0.842·9-s + 0.0158·11-s + 0.250·12-s − 0.839·13-s − 2.09·14-s − 1.23·16-s − 1.07·18-s + 1.79·19-s − 0.651·21-s + 0.0202·22-s + 0.217·23-s − 0.187·24-s − 1.07·26-s − 0.731·27-s − 1.03·28-s + 0.0781·29-s − 0.248·31-s − 1.10·32-s + 0.00628·33-s − 0.531·36-s + 0.0553·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.509331180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.509331180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 3 | \( 1 - 0.687T + 3T^{2} \) |
| 7 | \( 1 + 4.34T + 7T^{2} \) |
| 11 | \( 1 - 0.0525T + 11T^{2} \) |
| 13 | \( 1 + 3.02T + 13T^{2} \) |
| 19 | \( 1 - 7.82T + 19T^{2} \) |
| 23 | \( 1 - 1.04T + 23T^{2} \) |
| 29 | \( 1 - 0.420T + 29T^{2} \) |
| 31 | \( 1 + 1.38T + 31T^{2} \) |
| 37 | \( 1 - 0.336T + 37T^{2} \) |
| 41 | \( 1 - 6.59T + 41T^{2} \) |
| 43 | \( 1 - 9.99T + 43T^{2} \) |
| 47 | \( 1 - 6.13T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 5.09T + 59T^{2} \) |
| 61 | \( 1 - 5.97T + 61T^{2} \) |
| 67 | \( 1 - 0.916T + 67T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 + 5.39T + 73T^{2} \) |
| 79 | \( 1 - 9.98T + 79T^{2} \) |
| 83 | \( 1 + 6.53T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−7.61787666379416316769890958677, −7.12966482457434922899940870211, −6.29199683590787041215907992393, −5.66439529868498030479365948863, −5.25511231488685798725106324734, −4.15703381729486966049125480638, −3.56029886465478978526892860943, −2.81678986872520792936522796490, −2.54122981771872258263834657503, −0.60669862256767468776547789033,
0.60669862256767468776547789033, 2.54122981771872258263834657503, 2.81678986872520792936522796490, 3.56029886465478978526892860943, 4.15703381729486966049125480638, 5.25511231488685798725106324734, 5.66439529868498030479365948863, 6.29199683590787041215907992393, 7.12966482457434922899940870211, 7.61787666379416316769890958677
