L(s) = 1 | − 3-s − 2·4-s − 5-s + 1.73·7-s + 9-s + 2·12-s + 15-s + 4·16-s − 3.46·17-s − 5.19·19-s + 2·20-s − 1.73·21-s + 6·23-s + 25-s − 27-s − 3.46·28-s + 6.92·29-s + 31-s − 1.73·35-s − 2·36-s − 5·37-s + 3.46·41-s + 10.3·43-s − 45-s − 12·47-s − 4·48-s − 4·49-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.654·7-s + 0.333·9-s + 0.577·12-s + 0.258·15-s + 16-s − 0.840·17-s − 1.19·19-s + 0.447·20-s − 0.377·21-s + 1.25·23-s + 0.200·25-s − 0.192·27-s − 0.654·28-s + 1.28·29-s + 0.179·31-s − 0.292·35-s − 0.333·36-s − 0.821·37-s + 0.541·41-s + 1.58·43-s − 0.149·45-s − 1.75·47-s − 0.577·48-s − 0.571·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 5.19T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 1.73T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 13T + 97T^{2} \) |
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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
−8.715422139509778262017307101792, −8.312451801137936375363717875728, −7.30143486974116417653142659763, −6.45762518405874774990241488267, −5.47463473472307727305398745504, −4.55249069074579412970109114024, −4.30206529941477171525949680124, −2.91860150094527516812052659571, −1.34769574688992792030614946027, 0,
1.34769574688992792030614946027, 2.91860150094527516812052659571, 4.30206529941477171525949680124, 4.55249069074579412970109114024, 5.47463473472307727305398745504, 6.45762518405874774990241488267, 7.30143486974116417653142659763, 8.312451801137936375363717875728, 8.715422139509778262017307101792