| L(s) = 1 | + 2.33·2-s − 3.33·3-s + 3.42·4-s − 7.75·6-s + 1.09·7-s + 3.33·8-s + 8.08·9-s − 11.4·12-s − 5.33·13-s + 2.56·14-s + 0.900·16-s − 2.52·17-s + 18.8·18-s + 2.09·19-s − 3.66·21-s − 4.52·23-s − 11.0·24-s − 12.4·26-s − 16.9·27-s + 3.76·28-s + 1.57·29-s + 1.80·31-s − 4.56·32-s − 5.89·34-s + 27.7·36-s + 4.33·37-s + 4.89·38-s + ⋯ |
| L(s) = 1 | + 1.64·2-s − 1.92·3-s + 1.71·4-s − 3.16·6-s + 0.415·7-s + 1.17·8-s + 2.69·9-s − 3.29·12-s − 1.47·13-s + 0.684·14-s + 0.225·16-s − 0.613·17-s + 4.44·18-s + 0.481·19-s − 0.798·21-s − 0.944·23-s − 2.26·24-s − 2.43·26-s − 3.26·27-s + 0.712·28-s + 0.291·29-s + 0.323·31-s − 0.806·32-s − 1.01·34-s + 4.62·36-s + 0.711·37-s + 0.793·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.33T + 2T^{2} \) |
| 3 | \( 1 + 3.33T + 3T^{2} \) |
| 7 | \( 1 - 1.09T + 7T^{2} \) |
| 13 | \( 1 + 5.33T + 13T^{2} \) |
| 17 | \( 1 + 2.52T + 17T^{2} \) |
| 19 | \( 1 - 2.09T + 19T^{2} \) |
| 23 | \( 1 + 4.52T + 23T^{2} \) |
| 29 | \( 1 - 1.57T + 29T^{2} \) |
| 31 | \( 1 - 1.80T + 31T^{2} \) |
| 37 | \( 1 - 4.33T + 37T^{2} \) |
| 41 | \( 1 + 1.19T + 41T^{2} \) |
| 43 | \( 1 - 9.17T + 43T^{2} \) |
| 47 | \( 1 + 5.66T + 47T^{2} \) |
| 53 | \( 1 + 3.24T + 53T^{2} \) |
| 59 | \( 1 + 6.42T + 59T^{2} \) |
| 61 | \( 1 + 4.66T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 2.62T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 3.61T + 89T^{2} \) |
| 97 | \( 1 - 8.94T + 97T^{2} \) |
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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
−7.83509074359251535988694469870, −7.13557090726317441088559783286, −6.48517294209935754002965014394, −5.82438998505406899681502598588, −5.21981576335177316900999601974, −4.55758780829106875732677401037, −4.19421733119203839626740530248, −2.75616605756615398134406378290, −1.62259966048911427526674666697, 0,
1.62259966048911427526674666697, 2.75616605756615398134406378290, 4.19421733119203839626740530248, 4.55758780829106875732677401037, 5.21981576335177316900999601974, 5.82438998505406899681502598588, 6.48517294209935754002965014394, 7.13557090726317441088559783286, 7.83509074359251535988694469870
