| L(s) = 1 | + 2.82·3-s − 5-s − 2.35·7-s + 4.95·9-s − 0.877·11-s − 5.73·13-s − 2.82·15-s + 7.91·17-s − 2.90·19-s − 6.62·21-s − 23-s + 25-s + 5.51·27-s + 4.15·29-s − 1.44·31-s − 2.47·33-s + 2.35·35-s + 4.79·37-s − 16.1·39-s − 9.18·41-s − 9.52·43-s − 4.95·45-s + 5.19·47-s − 1.47·49-s + 22.3·51-s + 4.98·53-s + 0.877·55-s + ⋯ |
| L(s) = 1 | + 1.62·3-s − 0.447·5-s − 0.888·7-s + 1.65·9-s − 0.264·11-s − 1.59·13-s − 0.728·15-s + 1.92·17-s − 0.666·19-s − 1.44·21-s − 0.208·23-s + 0.200·25-s + 1.06·27-s + 0.772·29-s − 0.260·31-s − 0.430·33-s + 0.397·35-s + 0.788·37-s − 2.59·39-s − 1.43·41-s − 1.45·43-s − 0.738·45-s + 0.757·47-s − 0.210·49-s + 3.12·51-s + 0.685·53-s + 0.118·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 + 0.877T + 11T^{2} \) |
| 13 | \( 1 + 5.73T + 13T^{2} \) |
| 17 | \( 1 - 7.91T + 17T^{2} \) |
| 19 | \( 1 + 2.90T + 19T^{2} \) |
| 29 | \( 1 - 4.15T + 29T^{2} \) |
| 31 | \( 1 + 1.44T + 31T^{2} \) |
| 37 | \( 1 - 4.79T + 37T^{2} \) |
| 41 | \( 1 + 9.18T + 41T^{2} \) |
| 43 | \( 1 + 9.52T + 43T^{2} \) |
| 47 | \( 1 - 5.19T + 47T^{2} \) |
| 53 | \( 1 - 4.98T + 53T^{2} \) |
| 59 | \( 1 + 1.34T + 59T^{2} \) |
| 61 | \( 1 + 1.86T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 - 0.871T + 71T^{2} \) |
| 73 | \( 1 + 4.75T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 5.15T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 - 0.421T + 97T^{2} \) |
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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
−7.55414700006625663629398976310, −7.24871225142186963128470243965, −6.34327282788386100051232862734, −5.33336873884323451893081311285, −4.50998749885264271459883558986, −3.68813208964859560518665759508, −3.02572999530785141896850307939, −2.61568085936233533541815599037, −1.51375623178117382283586561038, 0,
1.51375623178117382283586561038, 2.61568085936233533541815599037, 3.02572999530785141896850307939, 3.68813208964859560518665759508, 4.50998749885264271459883558986, 5.33336873884323451893081311285, 6.34327282788386100051232862734, 7.24871225142186963128470243965, 7.55414700006625663629398976310
