| L(s) = 1 | + 3-s − 5-s − 3.77·7-s + 9-s + 3.77·11-s − 13-s − 15-s + 3.77·17-s − 6·19-s − 3.77·21-s + 1.77·23-s + 25-s + 27-s + 2·29-s − 6·31-s + 3.77·33-s + 3.77·35-s − 3.77·37-s − 39-s − 0.227·41-s − 8·43-s − 45-s − 6·47-s + 7.22·49-s + 3.77·51-s + 3.77·53-s − 3.77·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.42·7-s + 0.333·9-s + 1.13·11-s − 0.277·13-s − 0.258·15-s + 0.914·17-s − 1.37·19-s − 0.823·21-s + 0.369·23-s + 0.200·25-s + 0.192·27-s + 0.371·29-s − 1.07·31-s + 0.656·33-s + 0.637·35-s − 0.620·37-s − 0.160·39-s − 0.0356·41-s − 1.21·43-s − 0.149·45-s − 0.875·47-s + 1.03·49-s + 0.528·51-s + 0.518·53-s − 0.508·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 1.77T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 3.77T + 37T^{2} \) |
| 41 | \( 1 + 0.227T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 3.77T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 3.77T + 61T^{2} \) |
| 67 | \( 1 - 9.54T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 9.77T + 79T^{2} \) |
| 83 | \( 1 - 9.54T + 83T^{2} \) |
| 89 | \( 1 + 3.77T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−8.451821205513468645895059605120, −7.54225440031815645200007228379, −6.72303939801228107178959323815, −6.36771577744242474578545926227, −5.21511118548580463748221551927, −4.08967018547719839752149541835, −3.55102124503169898742770912310, −2.79549209518248051236013226869, −1.52495331483662016007609593875, 0,
1.52495331483662016007609593875, 2.79549209518248051236013226869, 3.55102124503169898742770912310, 4.08967018547719839752149541835, 5.21511118548580463748221551927, 6.36771577744242474578545926227, 6.72303939801228107178959323815, 7.54225440031815645200007228379, 8.451821205513468645895059605120
