| L(s) = 1 | + 2.82·11-s − 6·17-s − 8.48·19-s − 5·25-s − 6·41-s + 8.48·43-s − 7·49-s + 14.1·59-s − 8.48·67-s + 2·73-s − 2.82·83-s − 18·89-s − 10·97-s − 19.7·107-s − 18·113-s + ⋯ |
| L(s) = 1 | + 0.852·11-s − 1.45·17-s − 1.94·19-s − 25-s − 0.937·41-s + 1.29·43-s − 49-s + 1.84·59-s − 1.03·67-s + 0.234·73-s − 0.310·83-s − 1.90·89-s − 1.01·97-s − 1.91·107-s − 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 8.48T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.671667032314195687508284180003, −7.990105176557125811354520330297, −6.81479645932986999992756426926, −6.48949243629100763541065749465, −5.52533214783076788697525970455, −4.33774279172323691271012062540, −3.97732929635683301388987292145, −2.57017772158623774332178907514, −1.68939449489349813322570948018, 0,
1.68939449489349813322570948018, 2.57017772158623774332178907514, 3.97732929635683301388987292145, 4.33774279172323691271012062540, 5.52533214783076788697525970455, 6.48949243629100763541065749465, 6.81479645932986999992756426926, 7.990105176557125811354520330297, 8.671667032314195687508284180003
