| L(s) = 1 | − 2·3-s − 5-s − 0.828·7-s + 9-s − 4.82·11-s + 2·13-s + 2·15-s − 2.82·17-s + 0.828·19-s + 1.65·21-s + 8.82·23-s + 25-s + 4·27-s − 29-s + 10.4·31-s + 9.65·33-s + 0.828·35-s − 8.48·37-s − 4·39-s − 6·41-s − 6·43-s − 45-s + 0.343·47-s − 6.31·49-s + 5.65·51-s − 7.65·53-s + 4.82·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s − 0.313·7-s + 0.333·9-s − 1.45·11-s + 0.554·13-s + 0.516·15-s − 0.685·17-s + 0.190·19-s + 0.361·21-s + 1.84·23-s + 0.200·25-s + 0.769·27-s − 0.185·29-s + 1.88·31-s + 1.68·33-s + 0.140·35-s − 1.39·37-s − 0.640·39-s − 0.937·41-s − 0.914·43-s − 0.149·45-s + 0.0500·47-s − 0.901·49-s + 0.792·51-s − 1.05·53-s + 0.651·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4855184927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4855184927\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 0.828T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 0.343T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 7.65T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 3.65T + 89T^{2} \) |
| 97 | \( 1 - 4.48T + 97T^{2} \) |
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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
−7.59932487812108149441765818527, −6.87783239425211696312712032527, −6.37127013998031901860124061861, −5.63453358879925384067071873644, −4.85777586287927416587578480845, −4.65462151378253281909667432828, −3.24646832875909124368767226650, −2.88042051635152943426666082430, −1.48863754557940030231728729723, −0.36353875978956403783882246030,
0.36353875978956403783882246030, 1.48863754557940030231728729723, 2.88042051635152943426666082430, 3.24646832875909124368767226650, 4.65462151378253281909667432828, 4.85777586287927416587578480845, 5.63453358879925384067071873644, 6.37127013998031901860124061861, 6.87783239425211696312712032527, 7.59932487812108149441765818527
