| L(s) = 1 | + 3.16·3-s + 1.18·5-s − 2.23·7-s + 7.01·9-s + 1.19·11-s − 0.642·13-s + 3.74·15-s + 0.682·17-s + 7.29·19-s − 7.06·21-s + 0.441·23-s − 3.59·25-s + 12.6·27-s − 2.24·29-s − 4.61·31-s + 3.78·33-s − 2.64·35-s + 6.35·37-s − 2.03·39-s + 2.44·41-s + 0.990·43-s + 8.30·45-s + 4.74·47-s − 2.01·49-s + 2.15·51-s − 7.94·53-s + 1.41·55-s + ⋯ |
| L(s) = 1 | + 1.82·3-s + 0.529·5-s − 0.843·7-s + 2.33·9-s + 0.360·11-s − 0.178·13-s + 0.967·15-s + 0.165·17-s + 1.67·19-s − 1.54·21-s + 0.0921·23-s − 0.719·25-s + 2.44·27-s − 0.417·29-s − 0.828·31-s + 0.659·33-s − 0.447·35-s + 1.04·37-s − 0.325·39-s + 0.382·41-s + 0.151·43-s + 1.23·45-s + 0.691·47-s − 0.288·49-s + 0.302·51-s − 1.09·53-s + 0.191·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.364740682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.364740682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 - 3.16T + 3T^{2} \) |
| 5 | \( 1 - 1.18T + 5T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 - 1.19T + 11T^{2} \) |
| 13 | \( 1 + 0.642T + 13T^{2} \) |
| 17 | \( 1 - 0.682T + 17T^{2} \) |
| 19 | \( 1 - 7.29T + 19T^{2} \) |
| 23 | \( 1 - 0.441T + 23T^{2} \) |
| 29 | \( 1 + 2.24T + 29T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 - 6.35T + 37T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 - 0.990T + 43T^{2} \) |
| 47 | \( 1 - 4.74T + 47T^{2} \) |
| 53 | \( 1 + 7.94T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 1.22T + 61T^{2} \) |
| 67 | \( 1 - 8.84T + 67T^{2} \) |
| 71 | \( 1 + 2.37T + 71T^{2} \) |
| 73 | \( 1 - 7.25T + 73T^{2} \) |
| 79 | \( 1 + 7.49T + 79T^{2} \) |
| 83 | \( 1 - 7.79T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−9.591896545630172305132355752971, −9.060805498313454812909960910118, −7.952401445706741872824365767207, −7.44099871537874430368647664163, −6.52225446092537214818385843192, −5.43972310456729059174659384298, −4.11120186152396947326924643395, −3.32430764246146669839623218856, −2.59312066799484741894494479899, −1.46095036590766252378723291797,
1.46095036590766252378723291797, 2.59312066799484741894494479899, 3.32430764246146669839623218856, 4.11120186152396947326924643395, 5.43972310456729059174659384298, 6.52225446092537214818385843192, 7.44099871537874430368647664163, 7.952401445706741872824365767207, 9.060805498313454812909960910118, 9.591896545630172305132355752971
