| L(s) = 1 | − 1.66·3-s − 3.44·5-s − 4.43·7-s − 0.230·9-s − 1.18·11-s − 3.49·13-s + 5.72·15-s − 2.17·17-s − 5.57·19-s + 7.37·21-s + 7.37·23-s + 6.85·25-s + 5.37·27-s − 4.21·29-s + 7.45·31-s + 1.97·33-s + 15.2·35-s − 5.18·37-s + 5.82·39-s − 1.01·41-s + 0.120·43-s + 0.792·45-s − 3.07·47-s + 12.6·49-s + 3.61·51-s − 2.80·53-s + 4.09·55-s + ⋯ |
| L(s) = 1 | − 0.960·3-s − 1.53·5-s − 1.67·7-s − 0.0767·9-s − 0.358·11-s − 0.969·13-s + 1.47·15-s − 0.526·17-s − 1.27·19-s + 1.60·21-s + 1.53·23-s + 1.37·25-s + 1.03·27-s − 0.783·29-s + 1.33·31-s + 0.344·33-s + 2.57·35-s − 0.852·37-s + 0.931·39-s − 0.159·41-s + 0.0183·43-s + 0.118·45-s − 0.448·47-s + 1.80·49-s + 0.506·51-s − 0.385·53-s + 0.551·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 1.66T + 3T^{2} \) |
| 5 | \( 1 + 3.44T + 5T^{2} \) |
| 7 | \( 1 + 4.43T + 7T^{2} \) |
| 11 | \( 1 + 1.18T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 + 2.17T + 17T^{2} \) |
| 19 | \( 1 + 5.57T + 19T^{2} \) |
| 23 | \( 1 - 7.37T + 23T^{2} \) |
| 29 | \( 1 + 4.21T + 29T^{2} \) |
| 31 | \( 1 - 7.45T + 31T^{2} \) |
| 37 | \( 1 + 5.18T + 37T^{2} \) |
| 41 | \( 1 + 1.01T + 41T^{2} \) |
| 43 | \( 1 - 0.120T + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 + 2.80T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 4.55T + 67T^{2} \) |
| 71 | \( 1 - 0.710T + 71T^{2} \) |
| 73 | \( 1 + 3.34T + 73T^{2} \) |
| 79 | \( 1 - 6.35T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 9.71T + 89T^{2} \) |
| 97 | \( 1 - 7.58T + 97T^{2} \) |
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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
−7.28128484050077051607780458813, −6.68786875439998055473041382010, −6.37348002488417893117458602419, −5.28740926188208359670702642319, −4.73413926166936622258748191757, −3.91142390530039976644906685615, −3.17687725804833473117995310593, −2.49003628405549794142777630449, −0.60147958183965052945995551813, 0,
0.60147958183965052945995551813, 2.49003628405549794142777630449, 3.17687725804833473117995310593, 3.91142390530039976644906685615, 4.73413926166936622258748191757, 5.28740926188208359670702642319, 6.37348002488417893117458602419, 6.68786875439998055473041382010, 7.28128484050077051607780458813
