| L(s) = 1 | + 1.41·3-s − 5-s + 7-s − 0.999·9-s − 11-s + 3.41·13-s − 1.41·15-s − 0.585·17-s + 1.41·21-s − 8.82·23-s + 25-s − 5.65·27-s − 0.828·29-s + 1.75·31-s − 1.41·33-s − 35-s + 1.17·37-s + 4.82·39-s + 4.24·41-s − 6·43-s + 0.999·45-s − 1.41·47-s + 49-s − 0.828·51-s + 2.82·53-s + 55-s + 7.89·59-s + ⋯ |
| L(s) = 1 | + 0.816·3-s − 0.447·5-s + 0.377·7-s − 0.333·9-s − 0.301·11-s + 0.946·13-s − 0.365·15-s − 0.142·17-s + 0.308·21-s − 1.84·23-s + 0.200·25-s − 1.08·27-s − 0.153·29-s + 0.315·31-s − 0.246·33-s − 0.169·35-s + 0.192·37-s + 0.773·39-s + 0.662·41-s − 0.914·43-s + 0.149·45-s − 0.206·47-s + 0.142·49-s − 0.116·51-s + 0.388·53-s + 0.134·55-s + 1.02·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 0.585T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 - 1.17T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 - 7.89T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 + 9.17T + 71T^{2} \) |
| 73 | \( 1 + 5.07T + 73T^{2} \) |
| 79 | \( 1 + 6.48T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 0.828T + 89T^{2} \) |
| 97 | \( 1 - 9.31T + 97T^{2} \) |
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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
−7.86429522413992100086022719699, −7.25440057369826625929804497520, −6.15733903225667370282227917971, −5.72417282278302535023954224944, −4.61131974055418490713782246209, −3.94329497334431158979933671750, −3.23115227841863998510804380165, −2.39533465421795861343525698221, −1.48573952000959920148489260638, 0,
1.48573952000959920148489260638, 2.39533465421795861343525698221, 3.23115227841863998510804380165, 3.94329497334431158979933671750, 4.61131974055418490713782246209, 5.72417282278302535023954224944, 6.15733903225667370282227917971, 7.25440057369826625929804497520, 7.86429522413992100086022719699
