| L(s) = 1 | − 2-s + 4-s − 2.85·5-s + 4.48·7-s − 8-s + 2.85·10-s − 0.198·11-s + 3.70·13-s − 4.48·14-s + 16-s + 0.618·17-s − 0.259·19-s − 2.85·20-s + 0.198·22-s + 3.16·25-s − 3.70·26-s + 4.48·28-s − 7.49·29-s − 2.80·31-s − 32-s − 0.618·34-s − 12.8·35-s + 5.90·37-s + 0.259·38-s + 2.85·40-s − 8.45·41-s − 10.0·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.27·5-s + 1.69·7-s − 0.353·8-s + 0.903·10-s − 0.0598·11-s + 1.02·13-s − 1.19·14-s + 0.250·16-s + 0.150·17-s − 0.0594·19-s − 0.638·20-s + 0.0422·22-s + 0.632·25-s − 0.727·26-s + 0.847·28-s − 1.39·29-s − 0.504·31-s − 0.176·32-s − 0.106·34-s − 2.16·35-s + 0.971·37-s + 0.0420·38-s + 0.451·40-s − 1.32·41-s − 1.53·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 - 4.48T + 7T^{2} \) |
| 11 | \( 1 + 0.198T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 - 0.618T + 17T^{2} \) |
| 19 | \( 1 + 0.259T + 19T^{2} \) |
| 29 | \( 1 + 7.49T + 29T^{2} \) |
| 31 | \( 1 + 2.80T + 31T^{2} \) |
| 37 | \( 1 - 5.90T + 37T^{2} \) |
| 41 | \( 1 + 8.45T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 9.81T + 47T^{2} \) |
| 53 | \( 1 - 3.93T + 53T^{2} \) |
| 59 | \( 1 - 9.81T + 59T^{2} \) |
| 61 | \( 1 + 5.11T + 61T^{2} \) |
| 67 | \( 1 + 7.35T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 9.62T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.46204256441771438438967018356, −7.12035120386385411424426331791, −5.95082835940021911136620396496, −5.35187712877508112145439732406, −4.39806999658469691425052322567, −3.92434008855975391120715211069, −3.03329753787495511002548276071, −1.84987429205907515547580413088, −1.22831759686391321046989529368, 0,
1.22831759686391321046989529368, 1.84987429205907515547580413088, 3.03329753787495511002548276071, 3.92434008855975391120715211069, 4.39806999658469691425052322567, 5.35187712877508112145439732406, 5.95082835940021911136620396496, 7.12035120386385411424426331791, 7.46204256441771438438967018356
