| L(s) = 1 | + 3-s + 5-s + 3.55·7-s + 9-s + 2.79·11-s + 3.73·13-s + 15-s − 7.83·17-s − 6.27·19-s + 3.55·21-s + 23-s + 25-s + 27-s + 2.75·29-s + 2.48·31-s + 2.79·33-s + 3.55·35-s − 1.55·37-s + 3.73·39-s + 5.78·41-s + 4.54·43-s + 45-s + 6.27·47-s + 5.61·49-s − 7.83·51-s + 8.89·53-s + 2.79·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.34·7-s + 0.333·9-s + 0.842·11-s + 1.03·13-s + 0.258·15-s − 1.89·17-s − 1.44·19-s + 0.774·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.512·29-s + 0.447·31-s + 0.486·33-s + 0.600·35-s − 0.255·37-s + 0.597·39-s + 0.904·41-s + 0.693·43-s + 0.149·45-s + 0.915·47-s + 0.801·49-s − 1.09·51-s + 1.22·53-s + 0.376·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.525140906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.525140906\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 3.55T + 7T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 + 7.83T + 17T^{2} \) |
| 19 | \( 1 + 6.27T + 19T^{2} \) |
| 29 | \( 1 - 2.75T + 29T^{2} \) |
| 31 | \( 1 - 2.48T + 31T^{2} \) |
| 37 | \( 1 + 1.55T + 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 - 4.54T + 43T^{2} \) |
| 47 | \( 1 - 6.27T + 47T^{2} \) |
| 53 | \( 1 - 8.89T + 53T^{2} \) |
| 59 | \( 1 + 3.31T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 5.52T + 67T^{2} \) |
| 71 | \( 1 + 6.34T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 9.62T + 79T^{2} \) |
| 83 | \( 1 + 5.83T + 83T^{2} \) |
| 89 | \( 1 - 0.390T + 89T^{2} \) |
| 97 | \( 1 - 0.961T + 97T^{2} \) |
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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
−8.483098213147976300606885694125, −7.51103894696946461323201062361, −6.62451499934927589042330337775, −6.21503811545605091209712778348, −5.14483537785684037614113119429, −4.26107632148302159849148033819, −3.99802951729788397343134551040, −2.53104956173697510306672800826, −1.97629630351130329186832231925, −1.05018193230655848617316374164,
1.05018193230655848617316374164, 1.97629630351130329186832231925, 2.53104956173697510306672800826, 3.99802951729788397343134551040, 4.26107632148302159849148033819, 5.14483537785684037614113119429, 6.21503811545605091209712778348, 6.62451499934927589042330337775, 7.51103894696946461323201062361, 8.483098213147976300606885694125
