| L(s) = 1 | − 1.77e5·3-s − 3.95e9·7-s + 3.13e10·9-s − 3.56e12·13-s − 2.11e14·19-s + 7.00e14·21-s + 2.38e15·25-s − 5.55e15·27-s + 1.15e16·31-s + 2.57e17·37-s + 6.31e17·39-s + 5.42e17·43-s + 1.17e19·49-s + 3.74e19·57-s − 6.73e19·61-s − 1.24e20·63-s − 5.28e18·67-s − 2.35e20·73-s − 4.22e20·75-s − 4.14e20·79-s + 9.84e20·81-s + 1.41e22·91-s − 2.04e21·93-s − 1.61e21·97-s − 2.23e22·103-s + 2.65e22·109-s − 4.55e22·111-s + ⋯ |
| L(s) = 1 | − 3-s − 1.99·7-s + 9-s − 1.98·13-s − 1.81·19-s + 1.99·21-s + 25-s − 27-s + 0.454·31-s + 1.44·37-s + 1.98·39-s + 0.583·43-s + 2.99·49-s + 1.81·57-s − 1.54·61-s − 1.99·63-s − 0.0432·67-s − 0.749·73-s − 75-s − 0.554·79-s + 81-s + 3.97·91-s − 0.454·93-s − 0.226·97-s − 1.61·103-s + 1.02·109-s − 1.44·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{23}{2})\) |
\(\approx\) |
\(0.4198502795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4198502795\) |
| \(L(12)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{11} T \) |
| good | 5 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 7 | \( 1 + 3954581662 T + p^{22} T^{2} \) |
| 11 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 13 | \( 1 + 3566264416198 T + p^{22} T^{2} \) |
| 17 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 19 | \( 1 + 211308066581014 T + p^{22} T^{2} \) |
| 23 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 29 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 31 | \( 1 - 11550498557326034 T + p^{22} T^{2} \) |
| 37 | \( 1 - 257128060064352074 T + p^{22} T^{2} \) |
| 41 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 43 | \( 1 - 542156431674983642 T + p^{22} T^{2} \) |
| 47 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 53 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 59 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 61 | \( 1 + 67394602624928417446 T + p^{22} T^{2} \) |
| 67 | \( 1 + 5287275662394476662 T + p^{22} T^{2} \) |
| 71 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 73 | \( 1 + \)\(23\!\cdots\!54\)\( T + p^{22} T^{2} \) |
| 79 | \( 1 + \)\(41\!\cdots\!42\)\( T + p^{22} T^{2} \) |
| 83 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 89 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 97 | \( 1 + \)\(16\!\cdots\!42\)\( T + p^{22} T^{2} \) |
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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
−15.08300452391555442329383753632, −12.94206587446852982621829917178, −12.30734778056573943992387186066, −10.46115834884643067682764971740, −9.495055578680427465794114551685, −7.07533525275456792818127483153, −6.11117536509720259025801482004, −4.44982103007515240217294508925, −2.63555913681574115763763425516, −0.38409757722798112004665573829,
0.38409757722798112004665573829, 2.63555913681574115763763425516, 4.44982103007515240217294508925, 6.11117536509720259025801482004, 7.07533525275456792818127483153, 9.495055578680427465794114551685, 10.46115834884643067682764971740, 12.30734778056573943992387186066, 12.94206587446852982621829917178, 15.08300452391555442329383753632
