| L(s) = 1 | − 2.25·2-s + 3.09·4-s + 5-s + 0.706·7-s − 2.46·8-s − 2.25·10-s + 2.33·11-s − 13-s − 1.59·14-s − 0.622·16-s − 6.04·17-s − 7.91·19-s + 3.09·20-s − 5.25·22-s + 8.95·23-s + 25-s + 2.25·26-s + 2.18·28-s + 2.40·29-s − 3.06·31-s + 6.33·32-s + 13.6·34-s + 0.706·35-s − 2.38·37-s + 17.8·38-s − 2.46·40-s + 9.45·41-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 1.54·4-s + 0.447·5-s + 0.267·7-s − 0.871·8-s − 0.713·10-s + 0.702·11-s − 0.277·13-s − 0.426·14-s − 0.155·16-s − 1.46·17-s − 1.81·19-s + 0.691·20-s − 1.12·22-s + 1.86·23-s + 0.200·25-s + 0.442·26-s + 0.412·28-s + 0.446·29-s − 0.550·31-s + 1.11·32-s + 2.34·34-s + 0.119·35-s − 0.391·37-s + 2.89·38-s − 0.389·40-s + 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.25T + 2T^{2} \) |
| 7 | \( 1 - 0.706T + 7T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 17 | \( 1 + 6.04T + 17T^{2} \) |
| 19 | \( 1 + 7.91T + 19T^{2} \) |
| 23 | \( 1 - 8.95T + 23T^{2} \) |
| 29 | \( 1 - 2.40T + 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 + 2.38T + 37T^{2} \) |
| 41 | \( 1 - 9.45T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 9.72T + 47T^{2} \) |
| 53 | \( 1 + 5.90T + 53T^{2} \) |
| 59 | \( 1 + 2.97T + 59T^{2} \) |
| 61 | \( 1 + 9.24T + 61T^{2} \) |
| 67 | \( 1 + 1.69T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4.88T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 0.521T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 2.96T + 97T^{2} \) |
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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
−8.105381454958274269663873910684, −7.14311401558402159723130353890, −6.68573824318323014543406813485, −6.08611725494201671320031709449, −4.81664254059541678018296999180, −4.24338239150205160447388994147, −2.80102490458673287187750956131, −2.05810384488673534611488940155, −1.23717982262840381434251560862, 0,
1.23717982262840381434251560862, 2.05810384488673534611488940155, 2.80102490458673287187750956131, 4.24338239150205160447388994147, 4.81664254059541678018296999180, 6.08611725494201671320031709449, 6.68573824318323014543406813485, 7.14311401558402159723130353890, 8.105381454958274269663873910684
