| L(s) = 1 | − 3-s + 5-s − 0.300·7-s + 9-s − 4.87·11-s − 3.19·13-s − 15-s − 4.37·17-s + 7.36·19-s + 0.300·21-s + 8.95·23-s + 25-s − 27-s + 8.23·29-s + 1.76·31-s + 4.87·33-s − 0.300·35-s − 1.44·37-s + 3.19·39-s − 4.18·41-s − 11.2·43-s + 45-s − 3.68·47-s − 6.90·49-s + 4.37·51-s + 12.7·53-s − 4.87·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.113·7-s + 0.333·9-s − 1.46·11-s − 0.886·13-s − 0.258·15-s − 1.06·17-s + 1.69·19-s + 0.0656·21-s + 1.86·23-s + 0.200·25-s − 0.192·27-s + 1.52·29-s + 0.316·31-s + 0.848·33-s − 0.0508·35-s − 0.236·37-s + 0.511·39-s − 0.653·41-s − 1.70·43-s + 0.149·45-s − 0.537·47-s − 0.987·49-s + 0.612·51-s + 1.75·53-s − 0.657·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 0.300T + 7T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 13 | \( 1 + 3.19T + 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 23 | \( 1 - 8.95T + 23T^{2} \) |
| 29 | \( 1 - 8.23T + 29T^{2} \) |
| 31 | \( 1 - 1.76T + 31T^{2} \) |
| 37 | \( 1 + 1.44T + 37T^{2} \) |
| 41 | \( 1 + 4.18T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 2.07T + 59T^{2} \) |
| 61 | \( 1 + 8.81T + 61T^{2} \) |
| 71 | \( 1 + 4.57T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 7.33T + 83T^{2} \) |
| 89 | \( 1 + 9.26T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.052574122205232044935808871375, −7.10603864099912883224521241934, −6.79731433901186742537280082444, −5.69734112087472889966375695005, −4.98292889731732890721537404827, −4.74499860961241544582926846339, −3.13131039281380981340023251620, −2.62524607125788629564206969133, −1.31576975207790599924972908937, 0,
1.31576975207790599924972908937, 2.62524607125788629564206969133, 3.13131039281380981340023251620, 4.74499860961241544582926846339, 4.98292889731732890721537404827, 5.69734112087472889966375695005, 6.79731433901186742537280082444, 7.10603864099912883224521241934, 8.052574122205232044935808871375
