| L(s) = 1 | + 2.22e8·3-s + 1.71e10·4-s − 1.51e12·5-s + 3.28e16·9-s − 5.05e17·11-s + 3.82e18·12-s − 3.36e20·15-s + 2.95e20·16-s − 2.59e22·20-s + 2.38e23·23-s + 1.70e24·25-s + 3.60e24·27-s + 4.47e25·31-s − 1.12e26·33-s + 5.64e26·36-s + 3.76e26·37-s − 8.68e27·44-s − 4.97e28·45-s − 5.84e27·47-s + 6.57e28·48-s + 5.41e28·49-s + 1.22e29·53-s + 7.64e29·55-s + 1.11e30·59-s − 5.78e30·60-s + 5.07e30·64-s − 2.15e31·67-s + ⋯ |
| L(s) = 1 | + 1.72·3-s + 4-s − 1.98·5-s + 1.97·9-s − 11-s + 1.72·12-s − 3.41·15-s + 16-s − 1.98·20-s + 1.69·23-s + 2.93·25-s + 1.67·27-s + 1.98·31-s − 1.72·33-s + 1.97·36-s + 0.824·37-s − 44-s − 3.90·45-s − 0.219·47-s + 1.72·48-s + 49-s + 0.598·53-s + 1.98·55-s + 0.879·59-s − 3.41·60-s + 64-s − 1.95·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{35}{2})\) |
\(\approx\) |
\(3.879728293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.879728293\) |
| \(L(18)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + p^{17} T \) |
| good | 2 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 3 | \( 1 - 222600715 T + p^{34} T^{2} \) |
| 5 | \( 1 + 1512594730201 T + p^{34} T^{2} \) |
| 7 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 13 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 17 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 19 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 23 | \( 1 - \)\(23\!\cdots\!95\)\( T + p^{34} T^{2} \) |
| 29 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 31 | \( 1 - \)\(44\!\cdots\!03\)\( T + p^{34} T^{2} \) |
| 37 | \( 1 - \)\(37\!\cdots\!75\)\( T + p^{34} T^{2} \) |
| 41 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 43 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 47 | \( 1 + \)\(58\!\cdots\!50\)\( T + p^{34} T^{2} \) |
| 53 | \( 1 - \)\(12\!\cdots\!10\)\( T + p^{34} T^{2} \) |
| 59 | \( 1 - \)\(11\!\cdots\!87\)\( T + p^{34} T^{2} \) |
| 61 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 67 | \( 1 + \)\(21\!\cdots\!05\)\( T + p^{34} T^{2} \) |
| 71 | \( 1 + \)\(58\!\cdots\!93\)\( T + p^{34} T^{2} \) |
| 73 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 79 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 83 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 89 | \( 1 - \)\(10\!\cdots\!83\)\( T + p^{34} T^{2} \) |
| 97 | \( 1 - \)\(79\!\cdots\!15\)\( T + p^{34} T^{2} \) |
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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
−13.09790272931460433526751338282, −11.81957422563515401745259363518, −10.52033079195537784302088110800, −8.630478822808556908301021799576, −7.78847743431748030297303611762, −7.10923698511907643413482974654, −4.45869706796777281772923477320, −3.19212307109227489286542416963, −2.66979274400107491083739071452, −0.932373073788343452102010325163,
0.932373073788343452102010325163, 2.66979274400107491083739071452, 3.19212307109227489286542416963, 4.45869706796777281772923477320, 7.10923698511907643413482974654, 7.78847743431748030297303611762, 8.630478822808556908301021799576, 10.52033079195537784302088110800, 11.81957422563515401745259363518, 13.09790272931460433526751338282
