| L(s) = 1 | + 3-s − 5-s + 4.82·7-s + 9-s + 11-s + 5.65·13-s − 15-s − 6.82·17-s + 1.17·19-s + 4.82·21-s + 4·23-s + 25-s + 27-s + 0.828·29-s + 33-s − 4.82·35-s + 0.343·37-s + 5.65·39-s − 0.828·41-s + 3.17·43-s − 45-s + 4·47-s + 16.3·49-s − 6.82·51-s − 13.3·53-s − 55-s + 1.17·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.82·7-s + 0.333·9-s + 0.301·11-s + 1.56·13-s − 0.258·15-s − 1.65·17-s + 0.268·19-s + 1.05·21-s + 0.834·23-s + 0.200·25-s + 0.192·27-s + 0.153·29-s + 0.174·33-s − 0.816·35-s + 0.0564·37-s + 0.905·39-s − 0.129·41-s + 0.483·43-s − 0.149·45-s + 0.583·47-s + 2.33·49-s − 0.956·51-s − 1.82·53-s − 0.134·55-s + 0.155·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.855379738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.855379738\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 4.82T + 7T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 + 0.828T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−8.712467036295104978643198968595, −8.257506283693475089293387976944, −7.50563407396158848205176707776, −6.72106798873757498080537612128, −5.73128040437547016483917302648, −4.62114061221701144897874031811, −4.26215113462449310436068329124, −3.18423974469110956922011270208, −1.98820282892135820732756833986, −1.15749125850269882240041475256,
1.15749125850269882240041475256, 1.98820282892135820732756833986, 3.18423974469110956922011270208, 4.26215113462449310436068329124, 4.62114061221701144897874031811, 5.73128040437547016483917302648, 6.72106798873757498080537612128, 7.50563407396158848205176707776, 8.257506283693475089293387976944, 8.712467036295104978643198968595
